Recent zbMATH articles in MSC 11Bhttps://zbmath.org/atom/cc/11B2024-04-02T17:33:48.828767ZWerkzeugExtension of Hoffman's combinatorial identity via specific zeta-like serieshttps://zbmath.org/1529.050092024-04-02T17:33:48.828767Z"Genčev, Marian"https://zbmath.org/authors/?q=ai:gencev.marianThe author generalizes an identity of \textit{M. E. Hoffman} [Pac. J. Math. 152, No. 2, 275--290 (1992; Zbl 0763.11037)] involving the Bernoulli numbers in several directions. This is motivated by computations of multiple zeta values. In each case, the identity is of the form where the left-hand side is a sum over compositions satisfying some condition and the right-hand side is a well-known combinatorial sequence or single sum. Several identities are presented. For instance, binomial coefficients, Euler numbers, Catalan numbers, and Stirling numbers of the first kind appear on the right-hand side.
Reviewer: Arvind Ayyer (Bangalore)Ranking trees based on global centrality measureshttps://zbmath.org/1529.050412024-04-02T17:33:48.828767Z"Barghi, Amir"https://zbmath.org/authors/?q=ai:rahnamai-barghi.amir|barghi.amir"DeFord, Daryl"https://zbmath.org/authors/?q=ai:deford.daryl-rSummary: Trees, or connected graphs with no cycles, are a commonly studied combinatorial family. When many natural metrics on networks are applied to the set of all trees of a fixed order, the extremal values are realized at the star and path graphs. In this paper, we prove inequalities for several global centrality measures, such as global closeness and betweenness centralities, and for graphical Stirling numbers of the first kind that interpolate these extremes. Moreover, we provide two algorithms that allow us to traverse the space of non-isomorphic trees of a fixed order, one towards the star graph of the same order and the other towards the path. Furthermore, we investigate the relationship between these global centrality measures on the one hand and the \((n-2)\)nd Stirling numbers of the first kind for small trees on the other hand, demonstrating a strong association between them, in particular with respect to the hierarchical structures obtained from applying our two interpolating algorithms. Based on our observations from these small trees, we prove general bounds that relate the \((n - 2)\)nd Stirling numbers of the first kind of trees of order \(n\) to these global centrality measures. Finally, we provide two related approaches to totally order the set of all non-isomorphic trees of fixed order. We show that the totally ordering obtained from one of these approaches is consistent with the hierarchical structure obtained from our two tree interpolation algorithms in addition to being one of the features to use for predicting the \((n - 2)\)nd Stirling numbers of the first kind for small trees.Hosoya index of thorny polymershttps://zbmath.org/1529.050472024-04-02T17:33:48.828767Z"Došlić, Tomislav"https://zbmath.org/authors/?q=ai:doslic.tomislav"Németh, László"https://zbmath.org/authors/?q=ai:nemeth.laszlo"Podrug, Luka"https://zbmath.org/authors/?q=ai:podrug.lukaSummary: A matching in a graph \(G\) is a collection of edges of \(G\) such that no two of them share a vertex. The number of all matchings in \(G\) is called its Hosoya index. In this paper, we compute Hosoya indices of several classes of unbranched polymers made of cycles of the same lengths arranged around a middle path and decorated by attaching to each vertex, a given number of pendent vertices or thorns. We establish linear recurrences satisfied by those numbers and obtain explicit formulas in terms of Fibonacci polynomials and their generalizations. Some possible directions of future research are also indicated.The \(m=2\) amplituhedron and the hypersimplexhttps://zbmath.org/1529.051622024-04-02T17:33:48.828767Z"Parisi, Matteo"https://zbmath.org/authors/?q=ai:parisi.matteo"Sherman-Bennett, Melissa"https://zbmath.org/authors/?q=ai:sherman-bennett.melissa-u"Williams, Lauren"https://zbmath.org/authors/?q=ai:williams.lauren-kSummary: The hypersimplex \(\Delta_{k+1},n\) is the image of the positive Grassmannian \(Gr^{\geq 0}_{k+1,n}\) under the moment map. It is a polytope of dimension \(n-1\) in \(\mathbb{R}^n\). Meanwhile, the amplituhedron \(\mathcal{A}^Z_{n,k,2}\) is the image of \(Gr^{\geq 0}_{k,n}\) under an amplituhedron map \(\widetilde{Z}\) induced by a positive matrix \(Z\). Introduced in the context of scattering amplitudes, it is not a polytope, and is a full dimensional subset of \(Gr_{k,k+2}\). Nevertheless, there seem to be remarkable connections between these two objects, as conjectured by Lukowski-Parisi-Williams (LPW) [\textit{T. Łukowski} et al., Int. Math. Res. Not. 2023, No. 3 (2023; \url{doi:10.1093/imrn/rnad010})]. We use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out positroid polytopes -- moment map images of positroid cells -- translate into sign conditions cutting out Grasstopes -- amplituhedron map images of positroid cells. Moreover, we subdivide the amplituhedron into chambers, just as the hypersimplex can be subdivided into simplices -- with both chambers and simplices enumerated by the Eulerian numbers. We use these properties to prove the main conjecture of (LPW): a collection of positroid polytopes is a tiling of the hypersimplex if and only if the collection of T-dual Grasstopes is a tiling of the amplituhedron \(\mathcal{A}^Z_{n,k,2}\) for all \(Z\). We also prove Arkani-Hamed-Thomas-Trnka's conjectural sign-flip characterization of \(\mathcal{A}^Z_{n,k,2}\) [\textit{N. Arkani-Hamed} et al., J. High Energy Phys. 2018, No. 1, Paper No. 16, 41 p. (2018; Zbl 1384.81130)].Combinatorics of Newell-Littlewood numbershttps://zbmath.org/1529.051632024-04-02T17:33:48.828767Z"Gao, Shiliang"https://zbmath.org/authors/?q=ai:gao.shiliang"Orelowitz, Gidon"https://zbmath.org/authors/?q=ai:orelowitz.gidon"Ressayre, Nicolas"https://zbmath.org/authors/?q=ai:ressayre.nicolas"Yong, Alexander"https://zbmath.org/authors/?q=ai:yong.alexanderSummary: We give an exposition of recent developments in the study of Newell-Littlewood numbers. These are the tensor product multiplicities of Weyl modules in the stable range. They are also the structure coefficients of the Koike-Terada basis of the ring of symmetric functions. Two types of combinatorial results are exhibited, those obtained combinatorially starting from the definition of the numbers, and those that also employ geometric and/or representation theoretic methods.Identities arising from binomial-like formulas involving divisors of numbershttps://zbmath.org/1529.110042024-04-02T17:33:48.828767Z"Gryszka, Karol"https://zbmath.org/authors/?q=ai:gryszka.karolThe main focus of the paper under review is to present a large number of identities involving \(\omega(n)\), the number of distinct prime divisors of \(n\). The identities included involve, the Fibonacci, Lucas, Stirling and Lah numbers, as well as the Pochhammer symbols. Often the identities are simpler when \(n\) is squarefree.
A small selection of the many identities are included here. Let \(L_k\) be the \(k\)th Lucas number, and let \(F_k\) be the \(k\)th Fibonacci number.
If \(n\) is squarefree then:
\[\sum_{d|n} \frac{L_{\omega(d)+1}}{\omega(d)+1} = \frac{L_{2\omega(n)+2}-1}{\omega(n)+1}, \]
\[\sum_{d|n} \frac{F_{\omega(d)+1}}{\omega(d)+1} = \frac{F_{2\omega(n)+2}}{\omega(n)+1}, \]
and
\[\sum_{d|n,d >1} \frac{L_{\omega(d)}}{\omega(d)} = \sum_{k=1}^{\omega(n)} \frac{L_{2k}-1}{k}. \]
These identities arise as special cases of more general identities.
A major role is played by the following theorem. Assume that \(n\) has as distinct prime factors \(p_1\), \(p_2, \dots p_{\omega(n)}\) and \(n=p_1^{a_1} \cdots p_k^{a_{\omega(n)}}\). Then
\[\prod_{i=1}^{\omega(n)} (x+a_i y) = \sum_{d|n} x^{\omega(n)-\omega(d)}y^{\omega(d)}.\]
Reviewer: Joshua Zelinsky (New Haven)Lucas non-Wieferich primes in arithmetic progressions and the \textit{abc} conjecturehttps://zbmath.org/1529.110052024-04-02T17:33:48.828767Z"Anitha, K."https://zbmath.org/authors/?q=ai:anitha.k"Mumtaj Fathima, I."https://zbmath.org/authors/?q=ai:fathima.i-mumtaj"Vijayalakshmi, A. R."https://zbmath.org/authors/?q=ai:vijayalakshmi.a-rSummary: We prove the lower bound for the number of Lucas non-Wieferich primes in arithmetic progressions. More precisely, for any given integer\( k\ge 2\), there are \(\gg \log x\) Lucas non-Wieferich primes \(p\le x\) such that \(p\equiv \pm 1\pmod k\), assuming the \(abc\) conjecture for number fields.On Kaprekar's junction numbershttps://zbmath.org/1529.110072024-04-02T17:33:48.828767Z"Alekseyev, Max A."https://zbmath.org/authors/?q=ai:alekseyev.max-a"Sloane, N. J. A."https://zbmath.org/authors/?q=ai:sloane.neil-j-aSummary: A base \(b\) junction number \(u\) has the property that there are at least two ways to write it as \(u=v+s(v)\), where \(s(v)\) is the sum of the digits in the expansion of the number \(v\) in base \(b\). For the base 10 case, Kaprekar in the 1950's and 1960's studied the problem of finding \(K(n)\), the smallest \(u\) such that the equation \(u=v+s(v)\) has exactly \(n\) solutions. He gave the values \(K(2)=101\), \(K(3)=10^{13}+1\), and conjectured that \(K(4)=10^{24}+102\). In 1966 \textit{A. Narasinga Rao} [Math. Stud. 34 (1966), 79--84 (1967; Zbl 0178.04403)] gave the upper bound \(10^{1111111111124}+102\) for \(K(5)\), as well as upper bounds for \(K(6)\), \(K(7)\), \(K(8)\), and \(K(16)\).
In the present work, we derive a set of recurrences which determine \(K(n)\) for any base \(b\) and in particular imply that these conjectured values of \(K(n)\) are correct. The key to our approach is an apparently new recurrence for \(F(u)\), the number of solutions to \(u=v+s(v)\). We illustrate our method by computing the values of \(K(n)\) for \(n\ge 16\) and bases \(b\le 10\), and show that for each base \(K(n)\) grows as a tower of height proportional to \(\log_2(n)\). Rather surprisingly, the values of \(K(n)\) for base 5 are determined by the classical Thue-Morse sequence, which leads us to define generalized Thue-Morse sequences for other bases.On the representation of the natural numbers by powers of the golden meanhttps://zbmath.org/1529.110082024-04-02T17:33:48.828767Z"Dekking, Michel"https://zbmath.org/authors/?q=ai:dekking.michel"van Loon, Ad"https://zbmath.org/authors/?q=ai:van-loon.adSummary: In a base phi representation, a natural number is written as a sum of powers of the golden mean \(\varphi\). There are many ways to do this. Well known is the standard representation, introduced by \textit{George Bergman} in 1957 [Math. Mag. 31, 98--110 (1957; Zbl 0079.01003)], where a unique representation is obtained by requiring that no consecutive powers, \(\varphi^n\) and \(\varphi^{n+1}\), occur in the representation. In this paper, we introduce a new representation by allowing that the powers \(\varphi^0\) and \(\varphi^1\) may occur at the same time, but no other consecutive powers. We then argue that this representation is much closer to the classical representation of the natural numbers by powers of an integer than Bergman's standard representation.Maximal density and the kappa values for the families \(\{a,a+1,2a+1,n\}\) and \(\{a,a+1,2a+1,3a+1,n\}\)https://zbmath.org/1529.110102024-04-02T17:33:48.828767Z"Pandey, Ram Krishna"https://zbmath.org/authors/?q=ai:pandey.ram-krishna|pandey.ram-krishna.1"Rai, Neha"https://zbmath.org/authors/?q=ai:rai.nehaSummary: Let \(M\) be a set of positive integers. We study the maximal density \(\mu(M)\) of the sets of nonnegative integers \(S\) whose elements do not differ by an element in \(M\). In 1973, \textit{D. G. Cantor} and \textit{B. Gordon} [J. Comb. Theory, Ser. A 14, 281--287 (1973; Zbl 0277.10043)] established a formula for \(\mu(M)\) for \(|M|\leq 2\). Since then, many researchers have worked upon the problem and found several partial results in the case \(|M|\geq 3\), including some results in the case when \(M\) is an infinite set.
In this paper, we study the maximal density problem for the families \(M=\{a,a+1,2a+1,n\}\) and \(M=\{a,a+1,2a+1,3a+1,n\}\), where \(a\) and \(n\) are positive integers and \(n\) is sufficiently large. In most of the cases, we find bounds for the parameter \textit{kappa}, denoted by \(\kappa(M)\), which actually serves as a lower bound for \(\mu(M)\). The parameter \(\kappa(M)\) has already got its importance due to its rich connection with problems such as the ``lonely runner conjecture'' in Diophantine approximation and coloring parameters such as ``circular coloring'' and ``fractional coloring'' in graph theory.
We also give some partial results for the general family \(M=\{a,a +1,2a+1,\dots,(s-2)a+1,n\}\), where \(s\geq 5\) and mention related problems in the remaining cases for future work.On the structure of sets in a residue class ring with the same representation functionhttps://zbmath.org/1529.110112024-04-02T17:33:48.828767Z"Chen, Shi-Qiang"https://zbmath.org/authors/?q=ai:chen.shiqiang"Wang, Rui-Jing"https://zbmath.org/authors/?q=ai:wang.ruijing"Yu, Wang-Xing"https://zbmath.org/authors/?q=ai:yu.wang-xingSummary: Let \(m\) be an integer with \(m \geq 2\). For a given set \(S \subseteq \mathbb{Z}_m\) and \(\overline{n} \in \mathbb{Z}_m\), let \(R_S(\overline{n})\) denote the number of solutions of the equation \(\overline{n} = \overline{s} + \overline{s^\prime}\) with unordered pair \((\overline{s}, \overline{s^\prime}) \in S \times S\) and \(\overline{s} \neq \overline{s^\prime}\). In this paper, we determine the structure of sets \(A\), \(B\) satisfying \(A \cup B = \mathbb{Z}_m, | A \cap B | = 2\) and \(R_A(\overline{n}) = R_B(\overline{n})\) for all \(\overline{n} \in \mathbb{Z}_m\), where \(m\) is even.On the density of bounded baseshttps://zbmath.org/1529.110122024-04-02T17:33:48.828767Z"Fang, Jin-Hui"https://zbmath.org/authors/?q=ai:fang.jinhui.1|fang.jinhuiSummary: For a nonempty set \(A\) of integers and an integer \(n\), let \(r_A(n)\) be the number of representations of \(n\) in the form \(n=a+a'\), where \(a\le a'\) and \(a, a'\in A\), and \(d_A(n)\) be the number of representations of \(n\) in the form \(n=a-a'\), where \(a, a'\in A\). The binary support of a positive integer \(n\) is defined as the subset \(S(n)\) of nonnegative integers consisting of the exponents in the binary expansion of \(n\), i.e., \(n=\sum_{i\in S(n)} 2^i\), \(S(-n)=-S(n)\) and \(S(0)=\emptyset \). For real number \(x\), let \(A(-x,x)\) be the number of elements \(a\in A\) with \(-x\le a\le x\). The famous Erdős-Turán Conjecture states that if \(A\) is a set of positive integers such that \(r_A(n)\geqslant 1\) for all sufficiently large \(n\), then \(\limsup_{n\rightarrow\infty}r_A(n)=\infty \). In 2004, Nešetřil and Serra initially introduced the notation of ``bounded'' property and confirmed the Erdős-Turán conjecture for a class of \textit{bounded} bases. They also proved that, there exists a set \(A\) of integers satisfying \(r_A(n)=1\) for all integers \(n\) and \(|S(x)\bigcup S(y)|\le 4|S(x+y)|\) for \(x,y\in A\). On the other hand, Nathanson proved that there exists a set \(A\) of integers such that \(r_A(n)=1\) for all integers \(n\) and \(2\log x/\log 5+c_1\le A(-x,x)\le 2\log x/\log 3+c_2\) for all \(x\ge 1\), where \(c_1,c_2\) are absolute constants.
In this paper, following these results, we prove that, there exists a set \(A\) of integers such that: \(r_A(n)=1\) for all integers \(n\) and \(d_A(n)=1\) for all positive integers \(n\), \(|S(x)\bigcup S(y)|\le 4|S(x+y)|\) for \(x,y\in A\) and \(A(-x,x) > (4/\log 5)\log\log x+c\) for all \(x\ge 1\), where \(c\) is an absolute constant. Furthermore, we also construct a family of arbitrarily spare such sets \(A\).A new class of minimal asymptotic baseshttps://zbmath.org/1529.110132024-04-02T17:33:48.828767Z"Nathanson, Melvyn B."https://zbmath.org/authors/?q=ai:nathanson.melvyn-bernardLet \(\mathbb N_0\) be the set of nonnegative integers and let \(h\) be a positive integer. Let \(A, A_1, \dots, A_h \subset \mathbb N_0\). The sumset and the \(h\)-fold sumset are, respectively, \[A_1 + \dots + A_h = \{a_1 + \dots + a_h \mid a_i \in A_i\} \quad \text{ and } \quad hA = \underbrace{A + \dots + A}_{h \text{ summands}}.\]
The set \(A\) is a basis of order \(h\) if \(hA = \mathbb N_0\), and \(A\) is an asymptotic basis of order \(h\) if \(hA\) contains all sufficiently large integers. An asymptotic basis of order \(h\) is minimal if no proper subset of \(A\) is an asymptotic basis of order \(h\).
Minimal asymptotic bases are extremal objects in additive number theory and are related to the conjecture of \textit{P. Erdős} and \textit{P. Turán} [J. Lond. Math. Soc. 16, 212--215 (1941; Zbl 0061.07301)] that the representation function of an asymptotic basis of order \(h\) must be unbounded. At this time, there are few explicit constructions of minimal asymptotic bases. \textit{M. B. Nathanson} [J. Number Theory 6, 324--333 (1974; Zbl 0287.10051); Acta Arith. 49, No. 5, 525--532 (1988; Zbl 0601.10040)] used a \(2\)-adic construction to produce the first examples of minimal asymptotic bases. This method was extended to \(g\)-adically defined sets by many authors (see [\textit{F. Chen} and \textit{Y. Chen}, Eur. J. Comb. 32, No. 8, 1329--1335 (2011; Zbl 1236.11014); \textit{Y.-G. Chen}, C. R., Math., Acad. Sci. Paris 350, No. 21--22, 933--935 (2012; Zbl 1279.11012); \textit{Y.-G. Chen} and \textit{M. Tang}, Acta Arith. 185, No. 3, 275--280 (2018; Zbl 1439.11032); \textit{X.-D. Jia}, in: Number theory: New York seminar, 1991-1995. New York, NY: Springer. 201--209 (1996; Zbl 0867.11007); \textit{X. Jia} and \textit{M. B. Nathanson}, Acta Arith. 52, No. 2, 95--101 (1989; Zbl 0683.10043); \textit{J. B. Lee}, Period. Math. Hung. 26, No. 3, 211--218 (1993; Zbl 0791.11007); \textit{J. Li} and \textit{J. Li}, J. Math. Res. Appl. 36, No. 6, 651--658 (2016; Zbl 1374.11013); \textit{D. Ling} and \textit{M. Tang}, Bull. Aust. Math. Soc. 92, No. 3, 374--379 (2015; Zbl 1366.11016); \textit{D. Ling} and \textit{M. Tang}, Colloq. Math. 151, No. 1, 9--18 (2018; Zbl 1402.11015); \textit{C.-F. Sun}, Indag. Math., New Ser. 30, No. 1, 128--135 (2019; Zbl 1420.11022); \textit{C.-F. Sun}, J. Number Theory 218, 152--160 (2021; Zbl 1460.11018); \textit{C.-F. Sun} and \textit{T.-T. Tao}, Int. J. Number Theory 15, No. 2, 389--406 (2019; Zbl 1435.11024)]).
This paper constructs a new class of minimal asymptotic bases.
A \(\mathcal G\)-adic sequence is a strictly increasing sequence of positive integers \(\mathcal G = (g_i)_{i=0}^{\infty}\) such that \(g_0 = 1\) and \(g_{i-1} \mid g_i\) for every \(i \ge 1\). Let \(d_i = \frac{g_i}{g_{i-1}} \in \mathbb N\), so that \(d_i \ge 2\) and \(g_i = d_1 d_2 \dots d_i\). For \(0 \le i < j\), we have \(\frac{g_{i+j}}{g_i} = d_{i+1} d_{i+2} \dots d_{i+j}\). In this sense, the usual \(g\)-adic representation, where \(g \ge 2\) is an integer, uses the \(\mathcal G\)-adic sequence \(\mathcal G = (g^i)_{i=0}^{\infty}\) with quotients \(d_i = g\) for every \(i \ge 1\). It is known [\textit{M. B. Nathanson}, Am. Math. Mon. 121, No. 1, 5--17 (2014; Zbl 1373.11006); Springer Proc. Math. Stat. 220, 255--267 (2017; Zbl 1418.11043)] that every positive integer \(n\) has a unique \(\mathcal G\)-adic representation \(n = \sum_{j \in F} x_jg_j\), where \(x_j \in [1,d_{j+1}-1]\) for every \(j \in F\).
Lemma 1. \(g_M \le n < g_{M+1}\) if and only if \(\max(F) = M\).
Let \(\varnothing \neq W \subset \mathbb N_0\), and let \(\mathcal F^*(W)\) be the set of all nonempty finite subsets of \(W\). Let \(\mathcal G = (g_i)_{i=0}^{\infty}\) be a \(\mathcal G\)-adic sequence. We define \[A_{\mathcal G}(W) = \left\{ \sum_{j \in F} x_jg_j \mid F \in \mathcal F^*(W) \text{ and } x_j \in [1,d_{j+1}-1] \right\} \subset \mathbb N.\] Note that \(0 \not\in A_{\mathcal G}(W)\) since \(\varnothing \not\in \mathcal F^*(W)\).
Let \(h \ge 2\). A partition of \(\mathbb N_0\) is a sequence \(\mathcal W = (W_i)_{i=0}^{h-1}\) of nonempty pairwise disjoint sets such that \(\mathbb N_0 = W_0 \cup W_1 \cup \dots \cup W_{h-1}\). This paper proves the following results.
Theorem 1. Let \(\mathcal W = (W_i)_{i=0}^{h-1}\) be a partition of \(\mathbb N_0\), and let \(\mathcal G = (g_i)_{i=0}^{\infty}\) be a \(\mathcal G\)-adic sequence. The set \(A_{\mathcal G}(\mathcal W) = \bigcup_{i=0}^{h-1} A_{\mathcal G}(W_i)\) is an asymptotic basis of order \(h\) with \(h\)-fold sumset \(hA_{\mathcal G}(\mathcal W) = \{n \in \mathbb N_0 \mid n \ge h\}\).
Theorem 2. Let \(h \ge 2\) and let \(\mathcal W = (W_i)_{i=0}^{h-1}\) be a partition of \(\mathbb N_0\). The set \(A = \{0\} \cup A_{\mathcal G}(\mathcal W)\) is a basis of order \(h\) but not a minimal asymptotic basis of order \(h\).
Theorem 3. Let \(h \ge 2\) and let \(t\) be an integer such that \(t \ge 1 + \frac{\log h}{\log 2}\). Let \(\mathcal W = (W_i)_{i=0}^{h-1}\) be a partition of \(\mathcal N_0\) such that, for every \(i \in [0,h-1]\), there is an infinite set \(\mathcal M_i\) of positive integers such that \([M_i-t+1,M_i] \subset W_i\) for every \(M_i \in \mathcal M_i\). Let \(\mathcal G = (g_i)_{i=0}^{\infty}\) be a \(\mathcal G\)-adic sequence. The set \(A_{\mathcal G}(\mathcal W) = \bigcup_{i=0}^{h-1} A_{\mathcal G}(W_i)\) is a minimal asymptotic basis of order \(h\).
Corollary 1. Let \(\mathcal W = W_0 \cup W_1\) be a partition of \(\mathbb N_0\) such that both \(W_0\) and \(W_1\) contain infinitely many pairs of consecutive integers. Let \(\mathcal G = (g_i)_{i=0}^{\infty}\) be a \(\mathcal G\)-adic sequence. The set \(A_{\mathcal G}(\mathcal W) = A_{\mathcal G}(W_0) \cup A_{\mathcal G}(W_1)\) is a minimal asymptotic basis of order \(2\).
The paper ends by proposing some open problems.
1. For \(A = \{0\} \cup A_{\mathcal G}(\mathcal W)\) as in Theorem 1, determine all integers \(a \in A\) such that \(A \backslash \{a\}\) is either an asymptotic basis of order \(h\) or a minimal asymptotic basis.
2. Let \(\mathcal G = (g_i)_{i=1}^{\infty}\) be a \(\mathcal G\)-adic sequence and let \(h \ge 2\). Construct partitions \(\mathcal W = (W_i)_{i=0}^{h-1}\) of \(\mathbb N_0\) such that \(A_{\mathcal G}(\mathcal W)\) either is or is not a minimal asymptotic basis of order \(h\).
3. Let \((d_i)_{i=1}^{\infty}\) be a sequence of \(2\)s and \(3\)s. Let \(\mathcal G = (g_i)_{i=0}^{\infty}\) be the \(\mathcal G\)-adic sequence defined by \(g_0 = 1\) and \(g_i = \prod_{j=1}^i d_j\) for \(i \ge 1\). Consider problem 2 with respect to this \(\mathcal G\)-adic sequence. Of particular interest are the infinitely many \(\mathcal G\)-adic sequences \(\mathcal G = (g_i)_{i=1}^{\infty}\) with quotients \(\{d_{2i-1}, d_{2i}\} = \{2,3\}\) for all \(i \in \mathbb N\).
For the entire collection see [Zbl 1506.11002].
Reviewer: Sávio Ribas (Belo Horizonte)Some remarks on problems of subset sumhttps://zbmath.org/1529.110142024-04-02T17:33:48.828767Z"Tang, Min"https://zbmath.org/authors/?q=ai:tang.min|tang.min.1"Xu, Hongwei"https://zbmath.org/authors/?q=ai:xu.hongweiSummary: Let \(A = \{ a_1 < a_2 < \cdots\}\) be a sequence of integers and let \(P(A) = \{\sum \varepsilon_i a_i : a_i \in A, \varepsilon_i = 0\) or \(1, \sum \varepsilon_i < \infty\}\). Burr posed the following question: Determine conditions on integer sequences \(B\) that imply either the existence or the non-existence of \(A\) for which \(P(A)\) is the set of all non-negative integers not in \(B\). In this paper, we focus on some problems of subset sum related to Burr's question.On higher dimensional arithmetic progressions in Meyer setshttps://zbmath.org/1529.110152024-04-02T17:33:48.828767Z"Klick, Anna"https://zbmath.org/authors/?q=ai:klick.anna"Strungaru, Nicolae"https://zbmath.org/authors/?q=ai:strungaru.nicolaeSummary: In this paper we study the existence of higher dimensional arithmetic progressions in Meyer sets. We show that the case when the ratios are linearly dependent over \(\mathbb{Z}\) is trivial and focus on arithmetic progressions for which the ratios are linearly independent. Given a Meyer set \(\Lambda\) and a fully Euclidean model set \(\varLambda(W)\) with the property that finitely many translates of \(\varLambda(W)\) cover \(\Lambda \), we prove that we can find higher dimensional arithmetic progressions of arbitrary length with \(k\) linearly independent ratios in \(\Lambda\) if and only if \(k\) is at most the rank of the \({\mathbb Z}\)-module generated by \((\varLambda(W))\). We use this result to characterize the Meyer sets that are subsets of fully Euclidean model sets.Recent progress in Hilbert cubes theoryhttps://zbmath.org/1529.110162024-04-02T17:33:48.828767Z"Hegyvári, Norbert"https://zbmath.org/authors/?q=ai:hegyvari.norbertA \textit{Hilbert cube} (in \(\mathbb{Z}\)) is a set of the form \(H(x,A)=\left\{x + \sum_{i\in S} a_i : S\subseteq A\right\}\) for \(x\in \mathbb{Z}\) and \(A\subseteq \mathbb{Z}\). If \(A\) is finite with \(|A|=d\), then \(d\) is the \textit{dimension} of the cube, while \(H\) is called an infinite Hilbert cube if \(A\) is infinite. If \(\dim(H)=d\), then trivially \(|H|\leq 2^d\), and \(H\) is said to be \textit{\(\Delta\)-degenerate} if \(|H|=2^{\Delta d}\). A classical result of (pre-Ramsey) Ramsey theory on the integers, due to Hilbert himself, is that if \(N\) is sufficiently large with respect to \(r\) and \(d\), then any \(r\)-coloring of \([N]:=\{1,2,\dots,N\}\) yields a monochromatic \(d\)-dimensional Hilbert cube.
This paper provides a pleasant survey of results on Hilbert cubes, first focusing on results related to density. One such example is a result of Nathanson, which says the if \(E\subseteq \mathbb{Z}\) and \(d(E)=\lim_{N\to \infty} |E\cap [N]|/N = 1\), then \(E\) contains an infinite Hilbert cube. The paper generalizes this result to the setting of \(\sigma\)-finite groups.
The next section focuses on Hilbert cubes that avoid a given set, for example a result of the author which says that given any infinite \(B\subseteq \mathbb{Z}\), there exists an infinite Hilbert cube \(H(x,A)\) with \(H(x,A)\cap B = \emptyset\) such that \(|A\cap [N]|\) is at least roughly \(\frac{\sqrt{N}}{\sqrt{|B\cap[N]|}\log^{1+o(1)} N}\) infinitely often.
The paper closes with a section on \(\Delta\)-degenerate Hilbert cubes, which includes some new results. In particular, defining a \(d\)-dimensional Hilbert cube \(H\) to be \(\Delta_{\leq}\)-degenerate if \(|H|\leq 2^{\Delta d}\), it is shown that the set
\[
\mathcal{H} = \{A\subseteq [n] : |A|=d, H(x,A) \text{ is }\Delta_{\leq}\text{-degenerate}\}
\]
satisfies \[n^{(1+o_d(1))\Delta d} \leq |\mathcal{H}| \leq n^{\Delta d} 3^{d^2}. \]
For the entire collection see [Zbl 1479.11005].
Reviewer: Alex Rice (Jackson)On several notions of complexity of polynomial progressionshttps://zbmath.org/1529.110172024-04-02T17:33:48.828767Z"Kuca, Borys"https://zbmath.org/authors/?q=ai:kuca.borysSummary: For a polynomial progression
\[
(x,\,x+P_1(y),\dots,\,x+P_t(y)),
\]
we define four notions of complexity: Host-Kra complexity, Weyl complexity, true complexity and algebraic complexity. The first two describe the smallest characteristic factor of the progression, the third refers to the smallest-degree Gowers norm controlling the progression, and the fourth concerns algebraic relations between terms of the progressions. We conjecture that these four notions are equivalent, which would give a purely algebraic criterion for determining the smallest Host-Kra factor or the smallest Gowers norm controlling a given progression. We prove this conjecture for all progressions whose terms only satisfy homogeneous algebraic relations and linear combinations thereof. This family of polynomial progressions includes, but is not limited to, arithmetic progressions, progressions with linearly independent polynomials \(P_1,\dots,P_t\) and progressions whose terms satisfy no quadratic relations. For progressions that satisfy only linear relations, such as
\[
(x,\, x+y^2,\, x+2y^2,\, x+y^3,\, x+2y^3),
\]
we derive several combinatorial and dynamical corollaries: first, an estimate for the count of such progressions in subsets of \(\mathbb{Z}/N\mathbb{Z}\) or totally ergodic dynamical systems; second, a lower bound for multiple recurrence; and third, a popular common difference result in \(\mathbb{Z}/N\mathbb{Z}\). Lastly, we show that Weyl complexity and algebraic complexity always agree, which gives a straightforward algebraic description of Weyl complexity.Weak hypergraph regularity and applications to geometric Ramsey theoryhttps://zbmath.org/1529.110182024-04-02T17:33:48.828767Z"Lyall, Neil"https://zbmath.org/authors/?q=ai:lyall.neil"Magyar, Ákos"https://zbmath.org/authors/?q=ai:magyar.akosSummary: Let \(\Delta =\Delta_1\times \ldots \times \Delta_d\subseteq \mathbb{R}^n\), where \(\mathbb{R}^n=\mathbb{R}^{n_1}\times \cdots \times \mathbb{R}^{n_d}\) with each \(\Delta_i\subseteq \mathbb{R}^{n_i}\) a non-degenerate simplex of \(n_i\) points. We prove that any set \(S\subseteq \mathbb{R}^n\), with \(n=n_1+\cdots +n_d\) of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of the configuration \(\Delta \). In particular any such set \(S\subseteq \mathbb{R}^{2d}\) contains a \(d\)-dimensional cube of side length \(\lambda \), for all \(\lambda \geq \lambda_0(S)\). We also prove analogous results with the underlying space being the integer lattice. The proof is based on a weak hypergraph regularity lemma and an associated counting lemma developed in the context of Euclidean spaces and the integer lattice.Inverse theorem for certain directional Gowers uniformity normshttps://zbmath.org/1529.110192024-04-02T17:33:48.828767Z"Milićević, Luka"https://zbmath.org/authors/?q=ai:milicevic.lukaSummary: Let \(G\) be a finite-dimensional vector space over a prime field \(\mathbb{F}_p\) with some subspaces \(H_1, \dots, H_k\). Let \(f: G\to\mathbb{C}\) be a function. Generalizing the notion of Gowers uniformity norms, Austin introduced directional Gowers uniformity norms of \(f\) over \((H_1, \dots, H_k)\) as
\[
\|f\|_{\mathsf{U}(H_1, \dots, H_k)}^{2^k} = \mathop{\mathbb{E}}_{x\in G, h_1\in H_1, \dots, h_k\in H_K} {\cdot\kern-.51 em \Delta}_{h_1}\dots\,{\cdot\kern-.51 em \Delta}_{h_k}f(x)
\]
where \({\cdot\kern-.51 em \Delta}_u f(x):=f(x+u)\overline{f(x)}\) is the discrete multiplicative derivative.
Suppose that \(G\) is a direct sum of subspaces \(G = U_1\oplus U_2\oplus\cdots\oplus U_k\). In this paper we prove the inverse theorem for the norm
\[
\|\cdot\|_{\mathsf{U}(U_1,\dots, U_k, G, \dots,G)},
\]
with \(\ell\) copies of \(G\) in the subscript, which is the simplest interesting unknown case of the inverse problem for the directional Gowers uniformity norms. Namely, writing \(\|\cdot\|_{\mathsf{U}}\) for the norm above, we show that if \(f: G\to\mathbb{C}\) is a function bounded by 1 in magnitude and obeying \(\|f\|_{\mathsf{U}}\geq c\), provided \(\ell < p\), one can find a polynomial \(\alpha: G\to\mathbb{F}_p\) of degree at most \(k+\ell-1\) and functions \(g_i:\oplus_{j\in[k]\setminus\{i\}} U_j\to\{z\in\mathbb{C}: |z|\leqslant 1\}\) for \(i\in[k]\) such that
\[
\left|\mathop{\mathbb{E}}_{x\in G}f(x)\omega^{\alpha(x)}\prod\limits_{i\in[k]} g_i(x_1,\dots, x_{i-1}, x_{i+1}, \dots, x_k)\right| \\
\geqslant\left(\exp^{(O_{p, k, \ell}(1))}(O_{p, k ,\ell}({c}^{-1}))\right)^{-1}.
\]
The proof relies on an approximation theorem for the cuboid-counting function that is proved using the inverse theorem for Freiman multi-homomorphisms.Finding solutions with distinct variables to systems of linear equations over \(\mathbb{F}_p\)https://zbmath.org/1529.110202024-04-02T17:33:48.828767Z"Sauermann, Lisa"https://zbmath.org/authors/?q=ai:sauermann.lisaSummary: Let us fix a prime \(p\) and a homogeneous system of \(m\) linear equations \(a_{j,1}x_1+\cdots +a_{j,k}x_k=0\) for \(j=1,\ldots ,m\) with coefficients \(a_{j,i}\in \mathbb{F}_p\). Suppose that \(k\ge 3m\), that \(a_{j,1}+\cdots +a_{j,k}=0\) for \(j=1,\dots ,m\) and that every \(m\times m\) minor of the \(m\times k\) matrix \((a_{j,i})_{j,i}\) is non-singular. Then we prove that for any (large) \(n\), any subset \(A\subseteq \mathbb{F}_p^n\) of size \(|A|> C\cdot \Gamma^n\) contains a solution \((x_1,\dots ,x_k)\in A^k\) to the given system of equations such that the vectors \(x_1,\dots ,x_k\in A\) are all distinct. Here, \(C\) and \(\Gamma\) are constants only depending on \(p\), \(m\) and \(k\) such that \(\Gamma <p\). The crucial point here is the condition for the vectors \(x_1,\dots ,x_k\) in the solution \((x_1,\dots ,x_k)\in A^k\) to be distinct. If we relax this condition and only demand that \(x_1,\dots ,x_k\) are not all equal, then the statement would follow easily from Tao's slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.A quantitative Erdős-Fuchs type result for multivariate linear formshttps://zbmath.org/1529.110212024-04-02T17:33:48.828767Z"Yan, Xiao-Hui"https://zbmath.org/authors/?q=ai:yan.xiaohui"Li, Ya-Li"https://zbmath.org/authors/?q=ai:li.yaliThis paper is a contribution around a classical result in analytic number theory due to \textit{P. Erdős} and \textit{W. H. J. Fuchs} [J. Lond. Math. Soc. 31, 67--73 (1956; Zbl 0070.04104)]. For an infinite sequence of positive integers \(A\), we denote by \(R(A,n)\) the number of solutions to the equation \(x+y\leq n\) for \(x,y\in A\). Erdős and Fuchs proved that for any given \(c>0\) the equality
\[
R(A,n)=cn+o\left(n^{1/4}\log(n)^{-1/2}\right)
\]
it is not true. This paper deals with a generalization of this result. Consider \(A\) to be an infinite sequence of positive integers. Write \(m = \{(k_1,m_1),\cdots,(k_l,m_l)\}\) be a set of integer tuples such that \( \gcd(m_1,m_2,\cdots,m_l) > 1\). Denote by \(R_m(A,n)\) the number of different solutions of the equation \(k_1(a_{1,1} +\cdots+a_{1,m_1} ) +\dots +k_l(a_{l,1} +\cdots+a_{l,m_l})\leq n\) with \(a_{i,j}\in A\). The main contribution of the paper (Theorem 1.4) gets similar results for this representation function, improving previous results due to Rué, Chen-Tang, and Horváth.
Reviewer: Juanjo Rué Perna (Barcelona)Counterexamples to a conjecture of Dombi in additive number theoryhttps://zbmath.org/1529.110222024-04-02T17:33:48.828767Z"Bell, J. P."https://zbmath.org/authors/?q=ai:bell.jason-p"Shallit, J."https://zbmath.org/authors/?q=ai:shallit.jeffrey-oSummary: We disprove a conjecture of \textit{G. Dombi} [Acta Arith. 103, No. 2, 137--146 (2002; Zbl 1014.11009)] from additive number theory. More precisely, we find examples of sets \(A \subset\mathbb{N}\) with the property that \(\mathbb{N} \setminus A\) is infinite, but the sequence \(n \to \vert \{(a, b, c) : n = a + b + c \; \text{and}\; a, b, c \in A\}\vert \), counting the number of 3-compositions using elements of \(A\) only, is strictly increasing.Casting light on shadow Somos sequenceshttps://zbmath.org/1529.110232024-04-02T17:33:48.828767Z"Hone, Andrew N. W."https://zbmath.org/authors/?q=ai:hone.andrew-n-wSummary: Recently Ovsienko and Tabachnikov considered extensions of Somos and Gale-Robinson sequences, defined over the algebra of dual numbers. Ovsienko used the same idea to construct so-called shadow sequences derived from other nonlinear recurrence relations exhibiting the Laurent phenomenon, with the original motivation being the hope that these examples should lead to an appropriate notion of a cluster superalgebra, incorporating Grassmann variables. Here, we present various explicit expressions for the shadow of Somos-4 sequences and describe the solution of a general Somos-4 recurrence defined over the \(\mathbb{C}\)-algebra of dual numbers from several different viewpoints: analytic formulae in terms of elliptic functions, linear difference equations, and Hankel determinants.Sums of Fibonacci numbers indexed by integer partshttps://zbmath.org/1529.110242024-04-02T17:33:48.828767Z"Campbell, John M."https://zbmath.org/authors/?q=ai:campbell.john-maxwellSummary: Consider the integer sequences (\(F_{\lfloor\sqrt{n}\rfloor}: n\in\mathbb{N}_0\)) and (\(F_{\lfloor\log 2n \rfloor}:n\in\mathbb{N}\)), letting \(\lfloor x
\rfloor\) denote the integer part of a nonnegative value \(x\), and where \(F_n\) denotes the \(n\)th Fibonacci number for a nonnegative integer \(n\). We apply an Abel-type summation lemma to prove explicit evaluations for \(\sum^m_{n=1} F_{\lfloor\sqrt{n}\rfloor}\) and \(\sum^m_{n=1}F_{\lfloor\log 2n \rfloor}\) for a natural number \(m\). We then apply this summation lemma to determine an analytical formula for \(\sum^m_{n=1}F_{\lfloor\frac{n}{s}\rfloor}\), lettings \(s\) denote a natural number parameter, and we demonstrate how our method may be applied to evaluate sums of the form \(\sum^m_{n=1}F_{\left\lfloor\frac{\sqrt[r]{n}}{s}\right\rfloor}\) for integers \(r \geq 2\) and \(s \geq 1\). We also consider the problem of evaluating finite sums of expressions of the form \(F_{\left\lfloor\log_2\frac{n}{s}\right\rfloor}\) for a natural number \(s\). Much of our work is closely connected with evaluations for Fibonacci sums of the form \(S(t, m) = \sum^m_{n=1} n^tF_n\), where \(t\) is a nonnegative integer.Recurrence relations for \(S\)-legal index difference sequenceshttps://zbmath.org/1529.110252024-04-02T17:33:48.828767Z"Dantas E. Moura, Guilherme Zeus"https://zbmath.org/authors/?q=ai:moura.guilherme-zeus-dantas-e"Keisling, Andrew"https://zbmath.org/authors/?q=ai:keisling.andrew"Lilly, Astrid"https://zbmath.org/authors/?q=ai:lilly.astrid"Mauro, Annika"https://zbmath.org/authors/?q=ai:mauro.annika"Miller, Steven J."https://zbmath.org/authors/?q=ai:miller.steven-j"Phang, Matthew"https://zbmath.org/authors/?q=ai:phang.matthew"Velazquez Iannuzzelli, Santiago"https://zbmath.org/authors/?q=ai:velazquez-iannuzzelli.santiagoSummary: Zeckendorf's Theorem implies that the Fibonacci number \(F_n\) is the smallest positive integer that cannot be written as a sum of nonconsecutive previous Fibonacci numbers. \textit{M. Catral} et al. [Fibonacci Q. 54, No. 4, 348--365 (2016; Zbl 1400.11037)] studied a variation of the Fibonacci sequence, the Fibonacci Quilt sequence: the plane is tiled using the Fibonacci spiral, and integers are assigned to the squares of the spiral such that each square contains the smallest positive integer that cannot be expressed as the sum of nonadjacent previous terms. This adjacency is essentially captured in the differences of the indices of each square: the \(i\)th and \(j\)th squares are adjacent if and only if \(|i - j|\in\{1, 3, 4\}\) or \(\{i, j\} = \{1, 3\}\).
We consider a generalization of this construction: given a set of positive integers \(S\), the \(S\)-legal index difference (\(S\)-LID) sequence \((a_n)^\infty_{n=1}\) is defined by letting \(a_n\) be the smallest positive integer that cannot be written as \(\sum_{\ell\in L} a_\ell\) for some set \(L \subset [n-1]\) with \(|i - j|\notin S\) for all \(i, j \in L\). We discuss our results governing the growth of \(S\)-LID sequences, as well as results proving that many families of sets \(S\) yield \(S\)-LID sequences that follow simple recurrence relations.Balancing polynomials, Fibonacci numbers and some new series for \(\pi\)https://zbmath.org/1529.110262024-04-02T17:33:48.828767Z"Frontczak, Robert"https://zbmath.org/authors/?q=ai:frontczak.robert"Prasad, Kalika"https://zbmath.org/authors/?q=ai:prasad.kalikaThe aim of the paper under review is to improve the work of \textit{D. Castellanos} [Fibonacci Q. 24, 70--82 (1986; Zbl 0601.10006); ibid. 27, No. 5, 424--438 (1989; Zbl 0689.10020], who used odd Fibonacci and odd squared Fibonacci numbers to obtain some impressive series for \(\pi\). The authors examined some infinite series regarding balancing and Lucas-balancing polynomials. They obtained several new infinite series for \(\pi\) in terms of Fibonacci and Lucas numbers. Additionally, they utilized MATLAB and numerically verified their results. The authors are hoping to continue their investigation in this area by examining balancing polynomials and their relationships to classical orthogonal polynomials.
Reviewer: Mohammad K. Azarian (Evansville)On the greatest common divisor of \(n\) and the \(n\)th Fibonacci number. IIhttps://zbmath.org/1529.110272024-04-02T17:33:48.828767Z"Jha, Abhishek"https://zbmath.org/authors/?q=ai:jha.abhishek-kumar"Sanna, Carlo"https://zbmath.org/authors/?q=ai:sanna.carloSummary: Let \(\mathcal{A}\) be the set of all integers of the form \(\gcd (n, F_n)\), where \(n\) is a positive integer and \(F_n\) denotes the \(n\)th Fibonacci number. Leonetti and Sanna proved that \(\mathcal{A}\) has natural density equal to zero, and asked for a more precise upper bound. We prove that
\[
\#\big(\mathcal{A} \cap [1, x]\big) \ll \frac{x \log \log \log x}{\log \log x}
\]
for all sufficiently large \(x\). In fact, we prove that a similar bound also holds when the sequence of Fibonacci numbers is replaced by a general nondegenerate Lucas sequence.
For Part I, see [\textit{P. Leonetti} and the second author, Rocky Mt. J. Math. 48, No. 4, 1191--1199 (2018; Zbl 1437.11025)].Repdigits as difference of two Pell or Pell-Lucas numbershttps://zbmath.org/1529.110282024-04-02T17:33:48.828767Z"Keskin, Refik"https://zbmath.org/authors/?q=ai:keskin.refik"Erduvan, Fatih"https://zbmath.org/authors/?q=ai:erduvan.fatihLet \((P_n)_{n\ge 0}\) and \((Q_n)_{n\ge 0}\) be the sequences of \textit{Pell} and \textit{Pell-Lucas} numbers defined, respectively, by the linear recurrence relations \(P_0=0, P_1=1\), \(P_{n+2}=2P_{n+1}+P_n\) and \(Q_0=2, Q_1=2\), \(Q_{n+2}=2Q_{n+1}+Q_n\) for all \(n\ge 0\). A \textit{repdigit} is a positive \(R\) that has only one digit it its decimal expansion, that is, \(R\) is of the form
\[
R=\frac{d\cdot (10^{k}-1)}{9},
\]
for some positive integers \((d,k)\), with \(1\le d\le 9\) and \(k\ge 1\).
In the paper under review, the authors study the Diophantine equations
\[
P_n-P_m=\frac{d\cdot (10^{k}-1)}{9} \quad \text{and}\quad Q_n-Q_m=\frac{d\cdot (10^{k}-1)}{9},
\]
in non-negative integers \((n,m,d,k)\), where \(1\le d\le 9\), \(k\ge 1\), and \(1\le m<n\). In other words, the find all repdigits that can be written as a difference of two Pell numbers and all repdigits that can can written as a difference of two Pell-Lucas numbers. In their main results, they prove that the largest repdigit which is a difference of two Pell numbers is \(99=169-70=P_7-P_6\) and the largest repdigit which is a difference of two Pell-Lucas numbers is \(444=478-34=Q_7-Q_4\).
To prove their main results, the authors use a clever combination of techniques in Diophantine number theory, the usual properties of the Pell and Pell-Lucas sequences, Baker's theory of non-zero lower bounds for linear forms in logarithms of algebraic numbers, and the reduction techniques involving the theory of continued fractions. All computations can be done with the aid of simple computer programs in \texttt{SageMath} or \texttt{Mathematica}.
Reviewer: Mahadi Ddamulira (Kampala)On special spacelike hybrid numbers with Fibonacci divisor number componentshttps://zbmath.org/1529.110292024-04-02T17:33:48.828767Z"Kızılateş, Can"https://zbmath.org/authors/?q=ai:kizilates.can"Kone, Tiekoro"https://zbmath.org/authors/?q=ai:kone.tiekoroSummary: Hybrid numbers, whose components are defined as real numbers, are a mixture of complex numbers, dual numbers and hyperbolic numbers. These structures are frequently used both in pure mathematics and in many areas of physics. In this paper, by the help of the Fibonacci divisor numbers, we introduce the Fibonacci divisor hybrid numbers that generalize the Fibonacci hybrid numbers defined by Szynal-Liana and Wloch. We obtain miscellaneous algebraic properties of the Fibonacci divisor hybrid numbers. We also give an application related to the Fibonacci divisor hybrid numbers in matrices. Finally, using the character of the Fibonacci divisor hybrid numbers, we show that these numbers are spacelike.Sums involving gibonacci polynomial squares: generalizationshttps://zbmath.org/1529.110302024-04-02T17:33:48.828767Z"Koshy, Thomas"https://zbmath.org/authors/?q=ai:koshy.thomasSummary: We explore generalized versions of four sums involving gibonacci polynomial squares.Additional sums involving gibonacci polynomial squareshttps://zbmath.org/1529.110312024-04-02T17:33:48.828767Z"Koshy, Thomas"https://zbmath.org/authors/?q=ai:koshy.thomasSummary: We investigate four infinite sums involving gibonacci polynomial squares and their numeric versions, and deduce their Pell versions.Additional sums involving Jacobsthal polynomial squareshttps://zbmath.org/1529.110322024-04-02T17:33:48.828767Z"Koshy, Thomas"https://zbmath.org/authors/?q=ai:koshy.thomasSummary: We explore the Jacobsthal versions of four infinite sums involving gibonacci polynomial squares.\(\phi\)-expansions of rationalshttps://zbmath.org/1529.110332024-04-02T17:33:48.828767Z"Leung, King Shun"https://zbmath.org/authors/?q=ai:leung.king-shunSummary: Let \(\phi\) denote the golden ratio \((\sqrt{5} + 1)/2\) and \(x\) be a positive rational number. We study how the \(\phi\)-expansion of \(x\) can be found using some known results on Fibonacci numbers. We characterize those numbers in \((0, 1)\) with finite \(\phi\)-expansions. If \(x\in\mathbb{Q}\cap (0, 1)\), we give a precise expression for its \(\phi\)-expansion. In this case, the computation involves only simple operations on integers.\(k\)-Fibonacci numbers and \(k\)-Lucas numbers in Beatty sequences generated by powers of metallic meanshttps://zbmath.org/1529.110342024-04-02T17:33:48.828767Z"Noppakaew, Passawan"https://zbmath.org/authors/?q=ai:noppakaew.passawan"Kanwarunyu, Pavita"https://zbmath.org/authors/?q=ai:kanwarunyu.pavita"Wanitchatchawan, Parit"https://zbmath.org/authors/?q=ai:wanitchatchawan.paritSummary: For each positive integer \(k\), denote the metallic mean \(\left(k+\sqrt{k^2+ 4}\right)/2\) by \(\alpha_k\). In this article, we give some new identities involving the \(k\)-Fibonacci numbers, the \(k\)-Lucas numbers, metallic means, the floor function, and fractional parts. We also provide some properties of the Beatty sequence \(B(\alpha^n_k)\) generated by \(\alpha^n_k\), where \(n\) is any positive integer. Then these properties are used to show connections between \(k\)-Fibonacci and \(k\)-Lucas numbers and the sequence \(B(\alpha^n_k)\).Generalized commutative quaternion polynomials of the Fibonacci typehttps://zbmath.org/1529.110352024-04-02T17:33:48.828767Z"Szynal-Liana, Anetta"https://zbmath.org/authors/?q=ai:szynal-liana.anetta"Włoch, Iwona"https://zbmath.org/authors/?q=ai:wloch.iwona"Liana, Mirosław"https://zbmath.org/authors/?q=ai:liana.miroslawSummary: Generalized commutative quaternions is a number system which generalizes elliptic, parabolic and hyperbolic quaternions, bicomplex numbers, complex hyperbolic numbers and hyperbolic complex numbers. In this paper we introduce and study generalized commutative quaternion polynomials of the Fibonacci type.Sums of powers of integers and generalized Stirling numbers of the second kindhttps://zbmath.org/1529.110362024-04-02T17:33:48.828767Z"Cereceda, José Luis"https://zbmath.org/authors/?q=ai:cereceda.jose-luisSummary: By applying the Newton-Gregory expansion to the polynomial associated with the sum of powers of integers \(S_k(n) = 1^k+ 2^k+\cdots+n^k\), we derive a couple of infinite families of explicit formulas for \(S_k(n)\). One of the families involves the \(r\)-Stirling numbers of the second kind \(\{\substack{k\\j}\}_r\), \(j=0, 1, \ldots, k\), while the other involves their duals \(\{\substack{k\\j}\}_{-r}\), with both families of formulas being indexed by the non-negative integer \(r\). As a by-product, we obtain three additional formulas for \(S_k(n)\) involving the numbers \(\{\substack{k\\j}\}_{n+m}, \{\substack{k\\j}\}_{n-m}\), and \(\{\substack{k\\j}\}_{k-j}\), where \(m\) is any given non-negative integer. Furthermore, we provide several formulas for the Bernoulli polynomials in terms of the generalized Stirling numbers of the second kind, the harmonic numbers, and the so-called harmonic polynomials.Boole-Dunkl polynomials and generalizationshttps://zbmath.org/1529.110372024-04-02T17:33:48.828767Z"Asensi, Alejandro Gil"https://zbmath.org/authors/?q=ai:asensi.alejandro-gil"Labarga, Edgar"https://zbmath.org/authors/?q=ai:labarga-varona.edgar"Mínguez Ceniceros, Judit"https://zbmath.org/authors/?q=ai:minguez-ceniceros.judit"Varona, Juan Luis"https://zbmath.org/authors/?q=ai:varona-malumbres.juan-luisThe purpose of this paper is to study the Dunkl extension of the discrete Appell version of the generalized Euler polynomials, usually known as Euler polynomials of the second kind or Boole polynomials. After introducing a notation \(M_x\) for the mean operator, the generalized central Boole polynomials and the generalized central Euler polynomials of order \(r\) are defined by generating functions. Then the central difference operator is generalized to the Dunkl context by using a new kind of binomial coefficient called \(\mathrm{binomial}\,nk _{\alpha}\) and generalized Bernoulli-Dunkl and Euler-Dunkl polynomials of order \(r\) are defined by a generating function. This leads to the Stirling-Dunkl numbers of the first and second kind.
In order to define the Boole-Dunkl and generalized Boole-Dunkl polynomials, the Dunkl (central) mean operator is introduced. Several properties of Euler polynomials are generalized to the Dunkl context, and finally a Dunkl primitive of a function \(f\) together with a Dunkl integral is defined.
Reviewer: Thomas Ernst (Uppsala)Euler-Maclaurin summation formula on polytopes and expansions in multivariate Bernoulli polynomialshttps://zbmath.org/1529.110382024-04-02T17:33:48.828767Z"Brandolini, L."https://zbmath.org/authors/?q=ai:brandolini.luca"Colzani, L."https://zbmath.org/authors/?q=ai:colzani.leonardo"Gariboldi, B."https://zbmath.org/authors/?q=ai:gariboldi.bianca-m|gariboldi.bianca"Gigante, G."https://zbmath.org/authors/?q=ai:gigante.giacomo"Monguzzi, A."https://zbmath.org/authors/?q=ai:monguzzi.alessandroSummary: We provide a multidimensional weighted Euler-MacLaurin summation formula on polytopes and a multidimensional generalization of a result due to L. J. Mordell on the series expansion in Bernoulli polynomials. These results are consequences of a more general series expansion; namely, if \(\chi_{\tau\mathcal{P}}\) denotes the characteristic function of a dilated integer convex polytope \(\mathcal{P}\) and \(q\) is a function with suitable regularity, we prove that the periodization of \(q\chi_{\tau\mathcal{P}}\) admits an expansion in terms of multivariate Bernoulli polynomials. These multivariate polynomials are related to the Lerch Zeta function. In order to prove our results we need to carefully study the asymptotic expansion of \(\widehat{q\chi_{\tau\mathcal{P}}}\), the Fourier transform of \(q\chi_{\tau\mathcal{P}}\).Faulhaber polynomials and reciprocal Bernoulli polynomialshttps://zbmath.org/1529.110392024-04-02T17:33:48.828767Z"Kellner, Bernd C."https://zbmath.org/authors/?q=ai:kellner.bernd-cSummary: About four centuries ago, Johann Faulhaber developed formulas for the power sum \(1^n+2^n+\cdots+m^n\) in terms of \(m(m+1)/2\). The resulting polynomials are called the Faulhaber polynomials. We first give a short survey of Faulhaber's work and discuss the results of Jacobi (1834) and the less known ones of Schröder (1867), which already imply some results published afterwards. We then show, for suitable odd integers \(n\), the following properties of the Faulhaber polynomials \(F_n\). The recurrences between \(F_n, F_{n-1}\), and \(F_{n-2}\) can be described by a certain differential operator. Furthermore, we derive a recurrence formula for the coefficients of \(F_n\) that is the complement of a formula of Gessel and Viennot (1989). As a main result, we show that these coefficients can be expressed and computed in different ways by derivatives of generalized reciprocal Bernoulli polynomials, whose values can also be interpreted as central coefficients. This new approach finally leads to a simplified representation of the Faulhaber polynomials. As an application, we obtain some recurrences of the Bernoulli numbers, which are induced by symmetry properties.Some identities on generalized harmonic numbers and generalized harmonic functionshttps://zbmath.org/1529.110402024-04-02T17:33:48.828767Z"Kim, Dae San"https://zbmath.org/authors/?q=ai:kim.dae-san"Kim, Hyekyung"https://zbmath.org/authors/?q=ai:kim.hyekyung"Kim, Taekyun"https://zbmath.org/authors/?q=ai:kim.taekyunSummary: The harmonic numbers and generalized harmonic numbers appear frequently in many diverse areas such as combinatorial problems, many expressions involving special functions in analytic number theory, and analysis of algorithms. The aim of this article is to derive some identities involving generalized harmonic numbers and generalized harmonic functions from the beta functions \(F_n(x) = B(x+1, n+1)\), (\(n=0, 1, 2, \dots\)) using elementary methods. For instance, we show that the Hurwitz zeta function \(\zeta(x+1, r)\) and \(r!\) are expressed in terms of those numbers and functions, for every \(r = 2, 3, 4, 5\).On a family of Euler-type numbers and polynomialshttps://zbmath.org/1529.110412024-04-02T17:33:48.828767Z"Pita-Ruiz, Claudio"https://zbmath.org/authors/?q=ai:pita-ruiz.claudioSummary: We consider a family of Euler-type polynomials, depending on a real parameter \(\alpha \neq 0, 1\). The case \(\alpha = 2\) corresponds to standard Euler polynomials. We show some properties of these polynomials, and show also two generalized recurrences. As consequences of these results, we obtain several explicit formulas for Euler numbers and polynomials.Bell numbers and Kurepa's conjecturehttps://zbmath.org/1529.110422024-04-02T17:33:48.828767Z"Gallardo, Luis"https://zbmath.org/authors/?q=ai:gallardo.luis-hThe Kurepa conjecture [\textit{D. Kurepa}, Math. Balk. 1, 147--153 (1971; Zbl 0224.10009)] states
\[
0!+1!+\cdots+(p-1)!\not\equiv 0 \pmod p,
\]
for any odd prime number \(p\). In this paper, the author showed this conjecture holds under a mild condition for the set of prime numbers \(p\) such that \((\frac{p-1}{2})!=(\frac{2}{p})\) in \(\mathbb{F}_p\).
Reviewer: Toufik Mansour (Haifa)Combinatorial identities involving degenerate harmonic and hyperharmonic numbershttps://zbmath.org/1529.110432024-04-02T17:33:48.828767Z"Kim, Taekyun"https://zbmath.org/authors/?q=ai:kim.taekyun"Kim, Dae San"https://zbmath.org/authors/?q=ai:kim.dae-sanSummary: In recent years, some degenerate versions of quite a few special numbers and polynomials are introduced and investigated by means of various methods. The aim of this paper is to study some results on degenerate harmonic numbers, degenerate hyperharmonic numbers, degenerate Fubi polynomials and degenerate \(r\)-Fubini polynomials from a general identity which is valid for any two formal power series and involves the degenerate \(r\)-Stirling numbers of the second kind.New representations for all sporadic Apéry-like sequences, with applications to congruenceshttps://zbmath.org/1529.110442024-04-02T17:33:48.828767Z"Gorodetsky, Ofir"https://zbmath.org/authors/?q=ai:gorodetsky.ofirSummary: We find new representations, in terms of constant terms of powers of Laurent polynomials, for all the 15 sporadic Apéry-like sequences discovered by Zagier, \textit{G. Almkvist} and \textit{W. Zudilin} [AMS/IP Stud. Adv. Math. 38, 481--515 (2006; Zbl 1118.14043)] and \textit{S. Cooper} [Ramanujan J. 29, No. 1--3, 163--183 (2012; Zbl 1336.11031)]. The new representations lead to binomial expressions for the sequences, which, as opposed to previous expressions, do not involve powers of 3 or 8. We use these to establish the supercongruence \(B_{np^k} \equiv B_{np^{k-1}} \bmod{p^{2k}}\) for all primes \(p \geq 3\) and integers \(n, k \geq 1\), where \(B_n\) is a sequence discovered by Zagier, known as Sequence \(\mathbf{B}\). Additionally, for 14 of the 15 sequences, the Newton polytopes of the Laurent polynomials contain the origin as their only interior integral point. This property allows us to prove that these sequences satisfy a strong form of the Lucas congruences, extending work of Malik and Straub. Moreover, we obtain lower bounds on the \(p\)-adic valuation of these sequences via recent work of Delaygue.Some identities on type 2 degenerate Daehee polynomials and numbershttps://zbmath.org/1529.110452024-04-02T17:33:48.828767Z"Khan, Waseem A."https://zbmath.org/authors/?q=ai:khan.waseem-ahmad"Kamarujjama, M."https://zbmath.org/authors/?q=ai:kamarujjama.mohammadSummary: In this paper, we construct the type 2 degenerate Daehee numbers and polynomials and their higher-order analogues, and investigate some properties of these numbers and polynomials. In addition, we give some new identities and relations between the type 2 degenerate Daehee polynomials and degenerate Bernoulli polynomials of the second kind, an identity involving higher-order analogues of those polynomials and the degenerate Stirling numbers of the second kind, and an expression of higher-order analogues of those polynomials in terms of higher-order type 2 degenerate Bernoulli polynomials and the degenerate Stirling numbers of the first kind.On practical numbers of some special formshttps://zbmath.org/1529.110462024-04-02T17:33:48.828767Z"Wang, Li-Yuan"https://zbmath.org/authors/?q=ai:wang.liyuan"Sun, Zhi-Wei"https://zbmath.org/authors/?q=ai:sun.zhi-weiA positive integer \(m\) is called a practical number if each \(n= 1, \dots, m\) can be written as the sum of some distinct divisors of \(m\). In the paper under review, the authors study practical numbers by proving three results. First, they prove that for any integers \(b\geq 0\) and \(c>0\), if \(f(n):=n^2+bn+c\) is practical for some integer \(n>1\), then there are infinitely many non-negative integers \(n\) such that \(f(n)\) is practical. The second result asserts that \(2^{35\times 3^k+1}+2\) is practical for every integer \(k\geq 0\), hence implying that there are infinitely many practical numbers \(q\) with \(q^4 + 2\) also practical. Their third result is related by Pythagorean triples asserting that there are infinitely many practical Pythagorean triples \((a, b, c)\) with \(\gcd(a, b, c) = 4\) or \(6\).
Reviewer: Mehdi Hassani (Zanjan)The extended Frobenius problem for Fibonacci sequences incremented by a Fibonacci numberhttps://zbmath.org/1529.110482024-04-02T17:33:48.828767Z"Robles-Pérez, Aureliano M."https://zbmath.org/authors/?q=ai:robles-perez.aureliano-m"Rosales, José Carlos"https://zbmath.org/authors/?q=ai:rosales.jose-carlosSummary: We study the extended Frobenius problem for sequences of the form \(\{f_a+f_n\}_{n\in\mathbb{N}}\), where \(\{f_n\}_{n\in\mathbb{N}}\) is the Fibonacci sequence and \(f_a\) is a Fibonacci number. As a consequence of this study, we show that the family of numerical semigroups associated with these sequences satisfies Wilf's conjecture.Uniqueness conjecture on simultaneous Pell equationshttps://zbmath.org/1529.110492024-04-02T17:33:48.828767Z"Fujita, Yasutsugu"https://zbmath.org/authors/?q=ai:fujita.yasutsugu"Le, Maohua"https://zbmath.org/authors/?q=ai:le.maohuaConsider the simultaneous Pell equations
\[
x^2-(m^2-1)y^2=1, \quad z^2-(n^2-1)y^2=1, \quad x,y,z\in \mathbb{N},
\tag{1}
\]
for some integers \(m,n\) with \(1<m<n\). Obviously, the simultaneous Pell equations (1) have the solution \((x,y,z)=(m,1,n)\). In the paper under review, the authors consider the case when \(\gcd(m^2-1, n^2-1)\) is sufficiently large and prove the following theorems, which are the main results in the paper.
Theorem 1. Let \(a,b,c\) be positive integers with \(a<b\) and satisfying
\[
m^2-1=ac, \quad n^2-1=bc.
\tag{2}
\]
Assume that \(c\ge 5b^4\). Then the simultaneous Pell equations (1) have only the solution \((x,y,z)=(m,1,n)\).
Theorem 2. Let \(b,c\) be positive integers satisfying (2) with \(a=1\). Assume that \(b+1\) is a square and \(b-1\) is a prime power. Then the simultaneous Pell equations (1) have only the solution \((x,y,z)=(m,1,n)\).
To prove Theorem 1 and Theorem 2, the authors use a clever combination of techniques in number theory, Padé approximation method, Baker's method on linear forms in two logarithms, and the reduction techniques involving the theory of continued fractions.
Reviewer: Mahadi Ddamulira (Kampala)The elliptical case of an odds inversion problemhttps://zbmath.org/1529.110502024-04-02T17:33:48.828767Z"Hilmer, Kieran"https://zbmath.org/authors/?q=ai:hilmer.kieran"Jin, Angela"https://zbmath.org/authors/?q=ai:jin.angela"Lycan, Ron"https://zbmath.org/authors/?q=ai:lycan.ron"Ponomarenko, Vadim"https://zbmath.org/authors/?q=ai:ponomarenko.vadimSummary: A recent paper by \textit{R. K. Moniot} [Am. Math. Mon. 128, No. 2, 140--149 (2021; Zbl 1465.11081)] investigates the problem of, given a probability \(\frac{p}{q}\), finding a number of red and blue balls such that, when drawing two balls without replacement, the probability of drawing different colored balls is \(\frac{p}{q}\).
In this paper we deepen our understanding of the case where \(\frac{p}{q} > \frac12\) by finding bounds of the number of solutions for a given probability \(\frac{m}{2m-1}\) with \(m \in \mathbb{N}\) and characterize ``families'' of probabilities that are guaranteed to have more than two solutions. We also estimate the number of achievable probabilities in the ranges \(\bigl[\frac{m}{2m-1}, 1\bigr]\) and \(\bigl(\frac{m+1}{2m+1}, \frac{m}{2m-1}\bigr)\). Finally, we show that the ``recycling recurrence'' only exists for \(x_1 = n^2 - n\), \(y_1 = n^2\), and \(y_2 = n^2 + n\) for \(n \in \mathbb{N}\).On the ternary purely exponential Diophantine equation \((ak)^x + (bk)^y = ((a + b)k)^z\) for prime powers \(a\) and \(b\)https://zbmath.org/1529.110572024-04-02T17:33:48.828767Z"Le, Maohua"https://zbmath.org/authors/?q=ai:le.maohua"Soydan, Gökhan"https://zbmath.org/authors/?q=ai:soydan.gokhanLet \( k \) be a positive integer, and let \( a \) and \( b \) be coprime positive integers with \( a,b>1 \). In the paper under review, the authors completely study the Diophantine equation \begin{align*}
(ak)^x+(bk)^y=((a+b)k)^z, \quad (x,y,z)\in \mathbb{N}^3. \tag{1}
\end{align*}
Their main result is the following.
Theorem 1. Let \( r,s \) be positive integers, and let \( p,q \) be distinct odd primes. If \( k>1 \) and \( a,b \) satisfy one of the following conditions:
\begin{itemize}
\item[(i)] \( (a,b)=(2^{r}, p^{s}) \) with \( r>1 \); or
\item[(ii)] \( (a,b)=(p^{r}, 2^{s}) \) with \( s>1 \); or
\item[(iii)] \( (a,b)=(p^{r}, p^{s}) \),
\end{itemize}
then the Diophantine equation 1 has no solution \( (x,y,z)\in \mathbb{N}^3 \) with \( x>z>y \).
To prove Theorem 1, the authors use a clever combination of some elementary number theory techniques with classical results on the Nagell-Ljunggren equation, the Catalan equation, and some new properties of the classical Lucas sequence. Furthermore, Theorem 1 confirms the conjecture of \textit{P. Yuan} and \textit{Q. Han} in [Acta Arith. 184, No. 1, 37--49 (2018; Zbl 1410.11029)].
Reviewer: Mahadi Ddamulira (Kampala)Powers of Fibonacci numbers which are products of repdigitshttps://zbmath.org/1529.110582024-04-02T17:33:48.828767Z"Lourenço, Abel Medina"https://zbmath.org/authors/?q=ai:lourenco.abel-medinaLet \( \{F_n\}_{n\ge 0} \) be the classical sequence of \textit{Fibonacci numbers} defined by the linear recurrence relation: \( F_0=0 \), \( F_1=1 \), and \( F_{n+2}=F_{n+1}+F_n \) for all \( n\ge 0 \). In the paper under review, the author uses only elementary number theory methods, that is, divisibility properties, periodicity, results on prime factorizations, and a result on the Nagell-Ljunggren equation to show that the only solution to the Diophantine equation
\begin{align*}
F_n^{k}=\left(d_1\cdot \dfrac{10^{m}-1}{9}\right)\cdot\left(d_2\cdot \dfrac{10^{q}-1}{9}\right),
\end{align*}
in positive integers \( (n,k,d_1,d_2,m,q) \) with \( d_1,d_2\in \{1,2,\ldots, 9\} \), \( m, q\ge 2 \), and \( k\ge 2 \) is given by \( (n,k,d_1,d_2,m,q)=(10,2,5,5,2,2) \).
Reviewer: Mahadi Ddamulira (Kampala)Lucas generalized numbers in Narayana's cows sequencehttps://zbmath.org/1529.110592024-04-02T17:33:48.828767Z"Nikiema, Salifou"https://zbmath.org/authors/?q=ai:nikiema.salifou"Odjoumani, Japhet"https://zbmath.org/authors/?q=ai:odjoumani.japhetThe \textit{Narayana's cows sequence} \((N_m)_{m\ge 0}\) is defined by the ternary recurrence relation \(N_0=0\), \(N_1=N_2=1\), and \(N_{m+3}=N_{m+2}+N_m\) for all \(m\ge 0\). It is sequence \(A000930\) in the OEIS.
The \textit{generalized Lucas sequence} \((U_n)_{n\ge 0}\) with integer parameters \(a\ge 1\) and \(b=\pm 1\) is given by the binary recurrence relation \(U_0=0\), \(U_1=1\), and \(U_{n+2}=aU_{n+1}+bU_n\) for all \(n\ge 0\). When \((a,b)=(1,1), (2,1), (6,-1)\), this sequence coincides with the \textit{Fibonacci sequence}, \textit{Pell sequence}, and \textit{Balancing sequence}, respectively.
In the paper under review, the authors study the Diophantine equation
\[ N_m=U_n, \tag{1}\]
in positive integers \((m,n)\). Their main result is the following.
Theorem 1. Let \(n\) and \(m\) be positive integer solutions of the Diophantine equation (1). Then \(n\le m\). Moreover if \(m\ge 7\),
\[
m+2<1.546\cdot 10^{15}\left(\log\max\{\sqrt[3]{31}\sqrt{\Delta}, \alpha\}\right)^2\log\left(7.7281\cdot 10^{14}\left(\log\max\{\sqrt[3]{31}\sqrt{\Delta}, \alpha\}\right)^2\right),
\]
where \(\Delta:=a^2\pm 4\) and \(\alpha=\frac{a+\sqrt{\Delta}}{2}\).
Furthermore, as a consequence to Theorem 1, the authors prove that: the only Fibonacci numbers in Narayana's cows sequence are \(1,2,3,\) and \(13\); the only Pell numbers in Narayana's cows sequence are \(1\) and \(2\); and the only Balancing numbers in Narayana's cows sequence are \(1\) and \(6\).
To prove their main results, the authors use a clever combination of techniques in Diophantine number theory, the usual properties of the generalized Lucas sequence and the Narayana's cows sequence, Baker's theory of non-zero lower bounds for linear forms in logarithms of algebraic numbers, and the reduction techniques involving the theory of continued fractions. All computations can be done with the aid of simple computer programs in \texttt{SageMath} or \texttt{Mathematica}.
Reviewer: Mahadi Ddamulira (Kampala)On a variant of Pillai's problem with binary recurrences and \(S\)-unitshttps://zbmath.org/1529.110602024-04-02T17:33:48.828767Z"Ziegler, Volker"https://zbmath.org/authors/?q=ai:ziegler.volkerBased on the author's abstract: let \( U=(U_n)_{n\in \mathbb{N}} \) be a fixed binary recurrence sequence with real characteristic roots \( \alpha \) and \( \beta \) satisfying the dominant root condition, \( |\alpha|>|\beta| \), and let \( p_1,p_2,\ldots, p_s \) be a list of fixed distinct prime numbers.
In the paper under review, the author shows that there exist effectively computable constants \( C^+ \) and \( C^- \) such that the Diophantine equation
\[
U_n-p_1^{x_1}p_2^{x_2}\cdots p_s^{x_s}=c,
\]
has at most \( s \) solutions \( (n,x_1,x_2,\ldots,x_s)\in \mathbb{N}^{s+1} \) if \( c>C^+ \) and at most \( s+1 \) solutions \( (n,x_1,x_2,\ldots,x_s)\in \mathbb{N}^{s+1} \) if \( c<C^- \).
Furthermore, the author demonstrates the strength of this method by showing that for the binary recurrence sequence \( A=(A_n)_{n\in \mathbb{N}} \), with \( A_0=0 \), \( A_1=1 \) and \( A_{n+2}=3A_{n+1}+A_n \) for all \( n\ge 2 \), the Diophantine equation
\[
A_n-2^{x_1}3^{x_2}5^{x_3}17^{x_4}19^{x_5}=c,
\]
has at most six solutions \( (n,x_1,x_2,x_3,x_4,x_5) \) unless \( c=-35 \), in which case it has exactly seven solutions.
To prove the main results, the author uses a clever combination of techniques in number theory, the usual properties of binary recurrence sequences, Baker's theory for non-zero lower bounds for linear forms in complex and \( p \)-adic logarithms, and reduction techniques involving the theory of continued fractions, as well as the LLL algorithm. All computations are done with the aid of a computer program in \texttt{Mathematica}.
Reviewer: Mahadi Ddamulira (Kampala)Sums of Fibonacci numbers that are perfect powershttps://zbmath.org/1529.110612024-04-02T17:33:48.828767Z"Ziegler, Volker"https://zbmath.org/authors/?q=ai:ziegler.volkerLet \( \{F_n\}_{n\ge 0} \) be the usual sequence of \textit{Fibonacci numbers} defined by the linear recurrence relation: \( F_0=0 \), \( F_1=1 \), and \( F_{n+2}=F_{n+1}+F_n \) for all \( n\ge 0 \). In the paper under review, the author proves the following theorem, which is the main result in the paper.
Theorem 1. Let \( y>1 \) be a fixed integer, then there exists at most one solution \( (n,m,a)\in \mathbb{Z}^{3} \) to the Diophantine equation
\begin{align*}
F_n+F_m=y^{a}, \quad n>m>a, ~a>0,
\end{align*}
unless \( y=2,3,4,6,10 \). In the case \( y=2,3,4,6, \) or \( 10 \) all the solutions are listed below:
\begin{align*}
y&=2,~ (n,m,a)=(4,2,2), (5,4,3), (7,4,4);\\
y&=3, ~(n,m,a)=(3,2,1), (6,2,2);\\
y&=4, ~(n,m,a)=(4,2,1), (7,4,2);\\
y&-6, ~(n,m,a)=(5,2,1), (9,3,2);\\
y&=10, ~(n,m,a)=(6,3,1), (16,7,3).
\end{align*}
Furthermore, as a corollary to Theorem 1, the author proves that for a fixed prime \( p \), the Diophantine equation
\begin{align*}
F_n+F_m =p^{a}, \quad n>m>1,~a>0,
\end{align*}
has at most one solution \( (n,m,a)\in \mathbb{Z}^{3} \), unless \( p=2 \) or \( p=3 \).
To prove these results, the author uses a clever combination of techniques in Diophantine number theory, the usual properties of the Fibonacci sequence, Baker's theory of non-zero lower bounds for linear forms in logarithms of algebraic numbers, and reduction techniques involving the theory of continued fractions as well as the LLL algorithm. All numerical computations can be done with the help of simple computer programs in \texttt{Mathematica} or \texttt{SageMath}.
Reviewer: Mahadi Ddamulira (Kampala)On the equal sum and product problemhttps://zbmath.org/1529.110632024-04-02T17:33:48.828767Z"Zakarczemny, Maciej Szymon"https://zbmath.org/authors/?q=ai:zakarczemny.maciej-szymonFor a given positive integer \(n\), the present paper studies solutions to the equation
\[
x_1\cdots x_n=x_1+\cdots+x_n
\]
in positive integer variables \(x_1\leq x_2\leq\cdots\leq x_n\). The paper establishes, using only elementary methods, lower bounds on the number of solutions to the previous equation, either for individual values of \(n\) or on average over \(n\).
Particular attention is given to so-called \textit{exceptional values of the equal-sum-and-product-problem}, which are those values of \(n\) for which the previous equation only has one solution. It is conjectured [\textit{M. W. Ecker}, 75, No. 1, 41--47 (2002; \url{doi:10.2307/3219187})] that the set of such values of \(n\) is finite, and more specifically that it equals the set
\[
\{2,3,4,6,24,114,174,444\}.
\]
The main result of the paper, proved via explicit constructions, states that the desired number of solutions is at least
\[
\left\lfloor\frac{d(n-1)+1}{2}\right\rfloor+\left\lfloor\frac{d(2n-1)+1}{2}\right\rfloor-1
\]
where \(d\) denotes the divisor function. This places strong arithmetic restrictions on the set of exceptional values, and proves that they are somewhat rare, although it does not suffice to establish finiteness of the set.
Reviewer: Nuno Arala (Coventry)A study on a type of degenerate poly-Dedekind sumshttps://zbmath.org/1529.110662024-04-02T17:33:48.828767Z"Ma, Yuankui"https://zbmath.org/authors/?q=ai:ma.yuankui"Luo, Lingling"https://zbmath.org/authors/?q=ai:luo.lingling"Kim, Taekyun"https://zbmath.org/authors/?q=ai:kim.taekyun"Li, Hongze"https://zbmath.org/authors/?q=ai:li.hongze"Zhang, Wenpeng"https://zbmath.org/authors/?q=ai:zhang.wenpengSummary: Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations. As a new generalization of the Dedekind sums, the degenerate poly-Dedekind sums, which are obtained from the Dedekind sums by replacing Bernoulli functions by degenerate poly-Bernoulli functions of arbitrary indices are introduced in this article and are shown to satisfy a reciprocity relation.An extension of Maass theory to general Dirichlet serieshttps://zbmath.org/1529.110692024-04-02T17:33:48.828767Z"Wang, Xiao-Han"https://zbmath.org/authors/?q=ai:wang.xiaohan"Wang, Nian-Liang"https://zbmath.org/authors/?q=ai:wang.nianliang"Shigeru, Kanemitsu"https://zbmath.org/authors/?q=ai:shigeru.kanemitsuThe subject of this article is not only the Beta transform, Fourier-Whittaker expansion, Fourier-Bessel expansion, Maass form, Hardy transform, but also the modular relations for Maass forms associated with the Fourier-Whittaker expansion, Hecke Eisenstein series and also Epstein zeta functions. The authors also give some results within the scope of these topics, as well as tables and examples covering them. In addition, the authors provide comments to resolve the gap between the Hecke Eisenstein series and Epstein zeta functions, as well as the Chowla-Selberg integral formula and the Ramanujan-Guinand formula.
Reviewer: Yilmaz Simsek (Antalya)Bhargava's exponential functions and Bernoulli numbers associated to the set of prime numbershttps://zbmath.org/1529.110912024-04-02T17:33:48.828767Z"Adam, David"https://zbmath.org/authors/?q=ai:adam.david"Chabert, Jean-Luc"https://zbmath.org/authors/?q=ai:chabert.jean-lucSummary: Bhargava associates to each infinite subset \(E\) of \(\mathbb Z\) a sequence of positive integers called the factorials of \(E\). Such a sequence has many properties of the classical factorials, and Bhargava asked if the corresponding generalizations of functions defined by means of factorials could have interesting properties. In this paper we consider the generalization \(\exp_{}E(x)\) of the exponential function and the generalization \(B_{}E,n\) of the Bernoulli numbers. We are particularly interested in the case where the subset \(E\) is the set \(\mathbb P\) of prime numbers. We prove in particular that, for every rational \(r \neq - 2\) of the form \(\frac{\pm 1}{d}\) or \(\frac{\pm 2}{d}, \exp_{\mathbb P}(r)\) is irrational and that the Bernoulli polynomials without constant term \(B_{\mathbb P,n}(X)-B_{\mathbb P,n}(0)\) are integer polynomials. By the way, we answer a question from Mingarelli by showing that the sequence of factorials of the set \(\mathbb P\cup 2\mathbb P\) has infinitely many pairs of equal consecutive terms.
For the entire collection see [Zbl 1515.13002].Lochs-type theorems beyond positive entropyhttps://zbmath.org/1529.110942024-04-02T17:33:48.828767Z"Berthé, Valérie"https://zbmath.org/authors/?q=ai:berthe.valerie"Cesaratto, Eda"https://zbmath.org/authors/?q=ai:cesaratto.eda"Rotondo, Pablo"https://zbmath.org/authors/?q=ai:rotondo.pablo"Safe, Martín D."https://zbmath.org/authors/?q=ai:safe.martin-darioA classical theorem of \textit{G. Lochs} [Monatsh. Math. 67, 311--316 (1963; Zbl 0145.05002)] provides a probabilistic comparison between the relative speed of convergence of the decimal expansion and the continued fraction expansion of a number. In this paper, this is more generally interpreted as a relationship between partitions generated by numeration systems. This in turn allows the authors to derive Lochs-type theorems for rather general pairs of numeration systems. A novel feature of the approach is that positive entropy of the underlying partitions is no longer required.
To be more precise, consider a sequence of partitions \(\mathcal{P}\) of \([0,1]\), here meaning a sequence of disjoint collections of open intervals whose closure covers \([0,1]\), together with a Borel probability measure \(\lambda\) on \([0,1]\). Let \(f:\mathbb{N} \rightarrow\mathbb{R}\) be a function with \(\lim_{n \rightarrow \infty} f(n) = \infty\). The sequence of partitions is said to be log-balanced a.e. (resp. in measure) with respect to \(\lambda\) and with weight function \(f\) is
\[
\lim_{n \rightarrow \infty} \frac{-\log \lambda(I_n(x))}{f(n)} = 1 \quad \text{ a.e. (resp. in measure)} \quad (\lambda),
\]
where \(I_n(x)\) denotes the interval of the \(n\)'th partition of the family containing \(x\), which is well-defined as long as \(x\) is not an endpoint of a partition element. Now, given two such sequences of partitions, \(\mathcal{P}^1\) and \(\mathcal{P}^2\), and \(n \in \mathbb{N}\), define for \(x \in [0,1]\) not an endpoint of any partition element of either sequence of partitions, the Lochs' index,
\[
L_n(x, \mathcal{P}^1, \mathcal{P}^2) = \sup \{\ell \in \mathbb{N} : I^1_n(x) \subseteq I^2_{\ell}(c)\}.
\]
The original result of Lochs [loc. cit.] can be stated in terms of this index, when the families of partitions are those generated by the decimal expansion and the continued fraction expansion. Extending a result of \textit{K. Dajani} and \textit{A. Fieldsteel} [Proc. Am. Math. Soc. 129, No. 12, 3453--3460 (2001; Zbl 0999.11041)], the authors find general conditions under which
\[
\lim_{n \rightarrow \infty} \frac{f_2(L_n(x, \mathcal{P}^1, \mathcal{P}^2))}{f_1(n)}= 1,
\]
where the limit exists either \(\lambda\)-almost everywhere or in \(\lambda\)-measure, depending on whether the partitions are log-balanced almost everywhere or in measure.
In addition to the general Lochs-type results, the authors provide many examples of families of partitions satisfying one of the log-balanced conditions. For instance, any sequence of partitions of positive entropy \(h\) a.e. or in measure will be log-balanced with weight function \(f(n) = h n\). This includes base \(b\)-expansions, \(\beta\)-expansions and continued fractions. Additionally, results on log-balancedness of the Farey sequence of partitions, the Stern-Brocot sequence of partitions and the partitions generated by an irrational rotation of the circle are obtained. In each of these three cases, the weight function is sub-linear. The Farey sequence is log-balanced Lebesgue a.e., the Stern-Brocot sequence is log-balanced in Lebesgue measure but not a.e., and the rotation partitions are log-balanced Lebesgue a.e.for Lebesgue a.e.~rotation, although for uncountably many rotations the partitions are not even log-balanced in measure.
Reviewer: Simon Kristensen (Aarhus)Complement-finite idealshttps://zbmath.org/1529.130052024-04-02T17:33:48.828767Z"Baeth, N."https://zbmath.org/authors/?q=ai:baeth.nicholas-rSummary: We introduce a new class of commutative cancellative monoids which we call complement-finite ideals. Submonoids of free abelian monoids, these abstract monoids generalize the multiplicative structure of numerical monoids and are also closely related to monoids of zero-sum sequences and certain affine monoids. Here we provide a first general study of this more general construction. Specifically, we study the algebraic and arithmetic properties of complement-finite ideals and provide examples to illustrate their connections to other objects readily found in the literature.
For the entire collection see [Zbl 1515.13002].On negativity of Toeplitz-Hessenberg determinants whose elements contain large Schröder numbershttps://zbmath.org/1529.150062024-04-02T17:33:48.828767Z"Qi, Feng"https://zbmath.org/authors/?q=ai:qi.fengSummary: In the paper, by virtue of Wronski's formula and Kaluza's theorem for the power series and its reciprocal, and with the aid of the logarithmic convexity of a sequence constituted by the products of the factorials and the large Schröder numbers, the author presents negativity of a class of Toeplitz-Hessenberg determinants whose elements contain the products of the factorials and the large Schröder numbers.On the spectral norms of \(r\)-circulant and geometric circulant matrices with the bi-periodic hyper-Horadam sequencehttps://zbmath.org/1529.150162024-04-02T17:33:48.828767Z"Belaggoun, Nassima"https://zbmath.org/authors/?q=ai:belaggoun.nassima"Belbachir, Hacéne"https://zbmath.org/authors/?q=ai:belbachir.haceneSummary: In this paper, we define the bi-periodic hyper-Horadam sequence \(\{w_n^{(k)}\}_{n\in\mathbb{N}}\) and present its combinatorial properties. Moreover, we obtain upper and lower bounds for the spectral norms of different forms of the \(r\)-circulant and geometric circulant matrices with the bi-periodic hyper-Horadam sequence. Then we give some bounds for the spectral norms of the Kronecker and Hadamard products of these matrices.Description of \(J\)-sets and \(C\)-sets by matriceshttps://zbmath.org/1529.220012024-04-02T17:33:48.828767Z"Hosseini, Hedieh"https://zbmath.org/authors/?q=ai:hosseini.hedieh"Tootkaboni, Mohammad Akbari"https://zbmath.org/authors/?q=ai:tootkaboni.mohammad-akbariH. Furstenberg had used notions from topological dynamics and Ramsey theory, to define the so-called central subsets in semigroups, and prove a combinatorial result called the central sets theorem for commutative semigroups. This theorem combined van der Waerden's theorem and Hindman's theorem. \textit{D. De} et al. [Fundam. Math. 199, No. 2, 155--175 (2008; Zbl 1148.05052)] strengthened these results much further, and defined the concept of J-sets. Furstenberg's theorem as well as the generalizations by others use the set of finite subsets of sequences in the semigroup, to define \(J\)-sets. The present authors have obtained an interpretation which uses matrices instead of sequences. They show that their rephrasing produces the same \(J\)-sets. This matrix-theoretic interpretation enables them also to define a notion of \(C\)-sets that is related to the work of Hindman and Strauss on the algebra of Stone-Čech compactifications.
Reviewer: Balasubramanian Sury (Bangalore)Multi-solitons and integrability for a \((2+1)\)-dimensional variable coefficients Date-Jimbo-Kashiwara-Miwa equationhttps://zbmath.org/1529.354562024-04-02T17:33:48.828767Z"Zhao, Xue-Hui"https://zbmath.org/authors/?q=ai:zhao.xue-huiSummary: Under investigation in this paper is a \((2+1)\)-dimensional generalized variable coefficients Date-Jimbo-Kashiwara-Miwa equation, which describes the nonlinear dispersive wave in inhomogeneous media. Via the generalized Laurent series truncated at the constant-level term, an auto-Bäcklund transformation is derived. Bilinear forms, Bäcklund transformation, Lax pair and infinitely-many conservation laws are derived based on the binary Bell polynomials. Multi-soliton solutions are constructed via the Hirota method.Shrinking target horospherical equidistribution via translated Farey sequenceshttps://zbmath.org/1529.370012024-04-02T17:33:48.828767Z"Tseng, Jimmy"https://zbmath.org/authors/?q=ai:tseng.jimmySummary: For a certain diagonal flow on \(\mathrm{SL}(d, \mathbb{Z}) \setminus\mathrm{SL}(d, \mathbb{R})\) where \(d \geq 2\), we show that any bounded subset (with measure zero boundary) of the horosphere or a translated horosphere equidistributes, under a suitable normalization, on a target shrinking into the cusp. This type of equidistribution is \textit{shrinking target horospherical equidistribution (STHE)}, and we show STHE for several types of shrinking targets. Our STHE results extend known results for \(d = 2\) and \(\mathcal{L} \setminus \operatorname{PSL}(2, \mathbb{R})\) where \(\mathcal{L}\) is any cofinite Fuchsian group with at least one cusp. The two key tools needed to prove our STHE results for the horosphere are a renormalization technique and \textit{J. Marklof}'s result [Invent. Math. 181, No. 1, 179--207 (2010; Zbl 1200.11022)] on the equidistribution of the Farey sequence on distinguished sections. For our STHE results for translated horospheres, we introduce \textit{translated Farey sequences}, develop some of their geometric and dynamical properties, generalize Marklof's result by proving the equidistribution of translated Farey sequences for the same distinguished sections, and use this equidistribution of translated Farey sequences along with the renormalization technique to prove our STHE results for translated horospheres.Joint ergodicity of fractional powers of primeshttps://zbmath.org/1529.370032024-04-02T17:33:48.828767Z"Frantzikinakis, Nikos"https://zbmath.org/authors/?q=ai:frantzikinakis.nikosA finite family of sequences \(b_1, \dots , b_{\ell}: \mathbb{N} \rightarrow \mathbb{Z}\) is called jointly ergodic if for any ergodic dynamical system \((X,\mu, T)\) and functions \(f_1, \dots, f_{\ell} \in L^{\infty}(\mu)\), we have the following convergence in \(L^2(\mu)\):
\[
\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N T^{b_1(n)}f_1 \dots \cdot T^{b_{\ell}}f_{\ell}=\int f_1d\mu \dots \cdot \int f_{\ell}d\mu.
\]
The author of the present paper proved in [J. Anal. Math. 112, 79--135 (2010; Zbl 1211.37008)] that the pair of sequences \(\{[n^a], [n^{b}]\}\) with \(a, b\in {\mathbb{R}_+}\setminus \mathbb{Z}\) and \(a\neq b\), is jointly ergodic. By using the famous Furstenberg correspondence principle, one then concludes that any set of integers having positive upper density contains patterns of the form
\[
\{m, m+[n^a], m+[n^{b}]\},
\]
for some \(m,n\in\mathbb{N}\).
The present article aims at generalizing the result of the above mentioned paper in the case that the variable \(n\) is replaced by the \(n\)-th prime number \(p_n\). That is, he wants to prove that the pair of sequences \(\{[p_n^a], [p_n^{b}]\}\) with \(a, b\in {\mathbb{R}_+}\setminus \mathbb{Z}\) and \(a\neq b\), is jointly ergodic. More generally, we call \(a: \mathbb{R}_+ \rightarrow \mathbb{R}\) a fractional polynomial, if \(a(t)=\sum_{j=1}^r \alpha_j t^{d_j}\), where for \(1\leq j\leq r\), \(\alpha_j\in\mathbb{R}\) and \(d_j\in {\mathbb{R}_+\setminus \mathbb{Z}}\). The main theorem of the present article shows that for \(\ell\) linearly independent (over \(\mathbb{R}\)) fractional polynomials \(a_1,\dots, a_{\ell}\), the family of sequences \([a_1(p_n)], \dots, [a_{\ell}(p_n)]\) is jointly ergodic. Consequently, any set of integers having positive upper density contains patterns of the form
\[
\{m, m+[a_1(p_n)], \dots, m+[a_{\ell}(p_n)]\},
\]
for some \(m,n\in\mathbb{N}\).
Quite different to the classical method using Gowers uniformity norm of the modified von Mangoldt function, the proof of the main theorem is based on another result of the author in [Adv. Math. 417, Article ID 108918, 63 p. (2023; Zbl 1514.37019)] where it is proved that a finite family of sequences is jointly ergodic if and only if the sequences are ``good'' for equidistribution and seminorm estimates. The most part of the paper is devoted to the rather tricky proof of the seminorm estimates, while the equidistribution property follows from a theorem in [\textit{V. Bergelson} et al., Isr. J. Math. 201, Part B, 729--760 (2014; Zbl 1316.11062)].
Reviewer: Lingmin Liao (Créteil)Integral modification of beta-Apostol-Genocchi operatorshttps://zbmath.org/1529.410252024-04-02T17:33:48.828767Z"Neha"https://zbmath.org/authors/?q=ai:neha."Deo, Naokant"https://zbmath.org/authors/?q=ai:deo.naokantSummary: We propose certain Durrmeyer-type operators for Apostol-Genocchi polynomials in this research. We explore these operators' approximation attributes and measure the rate of convergence. In addition, we present a direct approximation theorem based on first and second-order modulus of continuity, local approximation findings for Lipschitz class functions and a direct theorem based on the typical modulus of continuity. Finally, we showed a graph illustrating the convergence of the suggested operators and an error table.Norm estimates for the Kakeya maximal function in high dimensions with respect to general measureshttps://zbmath.org/1529.420182024-04-02T17:33:48.828767Z"Gauvan, Anthony"https://zbmath.org/authors/?q=ai:gauvan.anthonySummary: We generalize Mitsis's work to higher dimensions and give application to a conditional estimate concerning Omega-Kakeya sets. Also, following Bourgain's arithmetic argument, we improve concrete bound on the Hausdorff dimension of an Omega-Kakeya set.Column-convex matrices, \(G\)-cyclic orders, and flow polytopeshttps://zbmath.org/1529.520102024-04-02T17:33:48.828767Z"González d'León, Rafael S."https://zbmath.org/authors/?q=ai:gonzalez-dleon.rafael-s"Hanusa, Christopher R. H."https://zbmath.org/authors/?q=ai:hanusa.christopher-r-h"Morales, Alejandro H."https://zbmath.org/authors/?q=ai:morales.alejandro-h"Yip, Martha"https://zbmath.org/authors/?q=ai:yip.marthaSummary: We study polytopes defined by inequalities of the form \(\sum_{i\in I}z_i\le 1\) for \(I\subseteq [d]\) and nonnegative \(z_i\) where the inequalities can be reordered into a matrix inequality involving a column-convex \(\{0,1\}\)-matrix. These generalize polytopes studied by Stanley, and the consecutive coordinate polytopes of Ayyer, Josuat-Vergès, and Ramassamy. We prove an integral equivalence between these polytopes and flow polytopes of directed acyclic graphs \(G\) with a Hamiltonian path, which we call spinal graphs. We show that the volumes of these flow polytopes are given by the number of upper (or lower) \(G\)-cyclic orders defined by the graphs \(G\). As a special case we recover results on volumes of consecutive coordinate polytopes. We study the combinatorics of \(k\)-Euler numbers, which are generalizations of the classical Euler numbers, and which arise as volumes of flow polytopes of a special family of spinal graphs. We show that their refinements, Ramassamy's \(k\)-Entringer numbers, can be realized as values of a Kostant partition function, satisfy a family of generalized boustrophedon recurrences, and are log concave along root directions. Finally, via our main integral equivalence and the known formula for the \(h^*\)-polynomial of consecutive coordinate polytopes, we give a combinatorial formula for the \(h^*\)-polynomial of flow polytopes of non-nested spinal graphs. For spinal graphs in general, we present a conjecture on upper and lower bounds for their \(h^*\)-polynomial.Topological gyrogroups with Fréchet-Urysohn property and \(\omega^{\omega}\)-basehttps://zbmath.org/1529.540162024-04-02T17:33:48.828767Z"Bao, Meng"https://zbmath.org/authors/?q=ai:bao.meng"Zhang, Xiaoyuan"https://zbmath.org/authors/?q=ai:zhang.xiaoyuan"Xu, Xiaoquan"https://zbmath.org/authors/?q=ai:xu.xiaoquanIn this work, the authors prove that a topological gyrogroup \(G\) is metrizable if and only if \(G\) has an \(\omega^{\omega}\)-base and \(G\) is Fréchet-Urysohn. Moreover, they prove that in topological gyrogroups, `every (countably, sequentially) compact subset being strictly (strongly) Fréchet-Urysohn' and `having an \(\omega^{\omega}\)-base' are all weakly three-space properties with \(H\) a closed L-subgyrogroup.
Reviewer: Watchareepan Atiponrat (Chiang Mai)Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equationhttps://zbmath.org/1529.650102024-04-02T17:33:48.828767Z"El-Sayed, Adel Abd Elaziz"https://zbmath.org/authors/?q=ai:el-sayed.adel-abd-elazizSummary: The nonlinear fractional-order cubic-quintic-heptic Duffing problem will be solved through a new numerical approximation technique. The suggested method is based on the Pell-Lucas polynomials' operational matrix in the fractional and integer orders. The studied problem will be transformed into a nonlinear system of algebraic equations. The numerical expansion containing unknown coefficients will be obtained numerically via applying Newton's iteration method to the claimed system. Convergence analysis and error estimates for the introduced process will be discussed. Numerical applications will be given to illustrate the applicability and accuracy of the proposed method.Larger corner-free sets from better NOF exactly-\(N\) protocolshttps://zbmath.org/1529.681082024-04-02T17:33:48.828767Z"Linial, Nati"https://zbmath.org/authors/?q=ai:linial.nathan"Shraibman, Adi"https://zbmath.org/authors/?q=ai:shraibman.adiSummary: A subset of the integer planar grid \([N] \times [N]\) is called \textit{corner-free} if it contains no triple of the form \((x,y), (x+\delta,y), (x,y+\delta)\). It is known that such a set has a vanishingly small density, but how large this density can be remains unknown. The best previous construction was based on Behrend's large subset of \([N]\) with no \(3\)-term arithmetic progression. Here we provide the first construction of a corner-free set that does not rely on a large set of integers with no arithmetic progressions. Our approach to the problem is based on the theory of communication complexity.
In the \(3\)-players exactly-\(N\) problem the players need to decide whether \(x+y+z=N\) for inputs \(x,y,z\) and fixed \(N\). This is the first problem considered in the multiplayer Number On the Forehead (NOF) model. Despite the basic nature of this problem, no progress has been made on it throughout the years. Only recently have explicit protocols been found for the first time, yet no improvement in complexity has been achieved to date. The present paper offers the first improved protocol for the exactly-\(N\) problem.Anticoncentration versus the number of subset sumshttps://zbmath.org/1529.681852024-04-02T17:33:48.828767Z"Jain, Vishesh"https://zbmath.org/authors/?q=ai:jain.vishesh"Sah, Ashwin"https://zbmath.org/authors/?q=ai:sah.ashwin"Sawhney, Mehtaab"https://zbmath.org/authors/?q=ai:sawhney.mehtaab-sSummary: Let \(\vec{w}=(w_1,\ldots,w_n)\in\mathbb{R}^n\). We show that for any \(n^{-2}\le\varepsilon\le 1\), if
\[
\#\{\vec{\xi}\in\{0,1\}^n:\langle\vec{\xi},\vec{w}\rangle=\tau\}\ge 2^{-\varepsilon n}\cdot 2^n
\]
for some \(\tau\in\mathbb{R}\), then
\[
\#\{\langle\vec{\xi},\vec{w}\rangle:\vec{\xi}\in\{0,1\}^n\}\le 2^{O(\sqrt{\varepsilon}n)}.
\]
This exponentially improves the \(\varepsilon\) dependence in a recent result of Nederlof, Pawlewicz, Swennenhuis, and Węgrzycki [Zbl 1529.68187; Zbl 1529.68186] and leads to a similar improvement in the parameterized (by the number of bins) runtime of bin packing.A faster exponential time algorithm for bin packing with a constant number of bins via additive combinatoricshttps://zbmath.org/1529.681862024-04-02T17:33:48.828767Z"Nederlof, Jesper"https://zbmath.org/authors/?q=ai:nederlof.jesper"Pawlewicz, Jakub"https://zbmath.org/authors/?q=ai:pawlewicz.jakub"Swennenhuis, Céline M. F."https://zbmath.org/authors/?q=ai:swennenhuis.celine-m-f"Węgrzycki, Karol"https://zbmath.org/authors/?q=ai:wegrzycki.karolSummary: In the Bin Packing problem one is given \(n\) items with weights \(w(1),\dots,w(n)\) and \(m\) bins with capacities \(c_1,\dots,c_m\). The goal is to partition the items into sets \(S_1,\dots,S_m\) such that \(w(S_j)\leqslant c_j\) for every bin \(j\), where \(w(X)\) denotes \(\sum_{i \in X}w(i)\). \textit{A. Björklund} et al. [SIAM J. Comput. 39, No. 2, 546--563 (2009; Zbl 1215.05056)] presented an \(\mathcal{O}^{\star}(2^n)\) time algorithm for Bin Packing (the \(\mathcal{O}^{\star}(\cdot)\) notation omits factors polynomial in the input size). In this paper, we show that for every \(m\in\mathbb{N}\) there exists a constant \(\sigma_m>0\) such that an instance of Bin Packing with \(m\) bins can be solved in \(\mathcal{O}(2^{(1-\sigma_m)n})\) randomized time. Before our work, such improved algorithms were not known even for \(m=4\). A key step in our approach is the following new result in Littlewood-Offord theory on the additive combinatorics of subset sums: For every \(\delta > 0\) there exists an \(\varepsilon > 0\) such that if \(|\{X\subseteq\{1,\dots,n\}:w(X)=v\}|\geqslant 2^{(1-\varepsilon)n}\) for some \(v\), then \(|\{ w(X):X\subseteq\{1,\dots,n\}\}|\leqslant 2^{\delta n}\).A faster exponential time algorithm for bin packing with a constant number of bins via additive combinatoricshttps://zbmath.org/1529.681872024-04-02T17:33:48.828767Z"Nederlof, Jesper"https://zbmath.org/authors/?q=ai:nederlof.jesper"Pawlewicz, Jakub"https://zbmath.org/authors/?q=ai:pawlewicz.jakub"Swennenhuis, Céline M. F."https://zbmath.org/authors/?q=ai:swennenhuis.celine-m-f"Węgrzycki, Karol"https://zbmath.org/authors/?q=ai:wegrzycki.karolAutomatic sequences of rank twohttps://zbmath.org/1529.682322024-04-02T17:33:48.828767Z"Bell, Jason P."https://zbmath.org/authors/?q=ai:bell.jason-p"Shallit, Jeffrey"https://zbmath.org/authors/?q=ai:shallit.jeffrey-oLet \(L\) be a language made of \(t\) finite words. If the right-infinite word \(\mathbf{x}\) belongs to \(L^\omega\), i.e., if \(\mathbf{x}\) is the concatenation of words in \(L\), then \(\mathbf{x}\) is said to be of rank \(t\). For instance, periodic words are of rank one.
The main result of this paper is to show that the property of being of rank two is decidable for automatic words. A key combinatorial component of this decision procedure is that there is a computable bound \(B\) such that, for each finite word \(y\), if \(y^B\) occurs as a factor of an automatic sequence \(\mathbf{x}\) then \(y\) occurs with unbounded exponent in \(\mathbf{x}\).
Reviewer: Michel Rigo (Liège)Congruence properties of combinatorial sequences via Walnut and the Rowland-Yassawi-Zeilberger automatonhttps://zbmath.org/1529.682392024-04-02T17:33:48.828767Z"Rampersad, Narad"https://zbmath.org/authors/?q=ai:rampersad.narad"Shallit, Jeffrey"https://zbmath.org/authors/?q=ai:shallit.jeffrey-oThe paper uses the Walnut software to prove a number of claims about congruence properties of Catalan numbers
\[
M_n = \frac 1 {n+1} \binom{2n}{n}\,,
\]
Motzkin numbers.
\[
C_n = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k} C_k\,,
\]
and central trinomial coefficients
\[
T_n = \sum_{k\geq 0}\binom{n}{2k}\binom{2k}{k}.
\]
Some of the proven results are new, some reprove old results in a new way. The proof method is uniform and its ingredients are as follows:
\begin{itemize}
\item The algorithm of \textit{E. Rowland} and \textit{D. Zeilberger} [J. Difference Equ. Appl. 20, No. 7, 973--988 (2014; Zbl 1358.68175)], which constructs a finite automaton computing the \(n\)-th element of the series \(\mathbf s_n \bmod p^\alpha\), where \(p\) is a prime, \(\alpha\) is a positive integer, and \(\mathbf s_n\) is a sequence of numbers that can be defined as constant terms of \(\left(P(x)\right)^nQ(x)\), where \(P\) and \(Q\) are Laurent polynomials (which is the case for all of the above sequences). The automaton computes the \(n\)-th element of the series by processing the base-\(k\) representation of \(n\). Sequences allowing such an automaton are called \(k\)-automatic.
\item The software Walnut, which is an implementation of a decision procedure for properties expressible by first-order formulas of automatic sequences.
\end{itemize}
The two ingredients above are taken as black boxes in the paper. As an example of a result to be found in the paper is that, for each number \(k\) whose base-5 expansion is of the form \(32^*\), a run of zeros starts in the sequence \(C_n \bmod 5\) (this is part of Theorem 13 which fully characterizes such runs of zeros).
Reviewer: Štěpán Holub (Praha)Well-posedness of the ambient metric equations and stability of even dimensional asymptotically de Sitter spacetimeshttps://zbmath.org/1529.830252024-04-02T17:33:48.828767Z"Kamiński, Wojciech"https://zbmath.org/authors/?q=ai:kaminski.wojciechSummary: The vanishing of the Fefferman-Graham obstruction tensor was used by \textit{M. T. Anderson} and \textit{P. T. Chruściel} [Commun. Math. Phys. 260, No. 3, 557--577 (2005; Zbl 1094.83002)] to show stability of the asymptotically de Sitter spaces in even dimensions. However, the existing proofs of hyperbolicity of this equation contain gaps. We show in this paper that it is indeed a well-posed hyperbolic system with unique up to diffeomorphism and conformal transformations smooth development for smooth Cauchy data. Our method applies also to equations defined by various versions of the Graham-Jenne-Mason-Sparling operators. In particular, we use one of these operators to propagate Gover's condition of being almost Einstein (basically conformal to Einsteinian metric). This allows us to study initial data also for Cauchy surfaces which cross the conformal boundary. As a by-product we show that on globally hyperbolic manifolds one can always choose a conformal factor such that Branson Q-curvature vanishes.