Recent zbMATH articles in MSC 11Jhttps://zbmath.org/atom/cc/11J2021-04-16T16:22:00+00:00WerkzeugDiscrepancy of generalized \(LS\)-sequences.https://zbmath.org/1456.111402021-04-16T16:22:00+00:00"Iacò, Maria Rita"https://zbmath.org/authors/?q=ai:iaco.maria-rita"Ziegler, Volker"https://zbmath.org/authors/?q=ai:ziegler.volkerIn this paper, the authors start with the classical sequences of points introduced by \textit{S. Kakutani} [Lect. Notes Math. 541, 369--375 (1976; Zbl 0363.60023)] using his splitting procedure. Later this procedure was generalized by \textit{I. Carbone} [Ann. Mat. Pura Appl. (4) 191, No. 4, 819--844 (2012; Zbl 1277.11080)] to $LS$-sequences and who proved discrepancy bound $k_1(\log N/N)$. The authors generalize $LS$-sequences for many parameters $L_1,\dots,L_k$, explicitly compute the constant $k_1$ and give discrepancy bound for a generalized $LS$-sequence. The authors also give an estimate of the star-discrepancy of the Kakutani-Fibonacci sequence. The paper continues work of \textit{M. Drmota} and \textit{M. Infusino} [Unif. Distrib. Theory 7, No. 1, 75--104 (2012; Zbl 1313.11084)].
Reviewer: Oto Strauch (Bratislava)Thue Diophantine equations.https://zbmath.org/1456.110342021-04-16T16:22:00+00:00"Waldschmidt, Michel"https://zbmath.org/authors/?q=ai:waldschmidt.michelThis survey presents the essentials concerning Thue (mainly) and Thue-Mahler equations. In this reviewer' s opinion, it is very much appropriate for graduate students of Mathematics looking for their way in Number Theory and for any mathematician who would like to know about what is about these equations and what kind of Mathematics are involved in their study.
Description of the paper's contents. Section 1: Basic definitions. The special case of Thue equation in which positive definite binary forms are involved, and CM-fields. General Thue equations and their relation to Diophantine Approximation. Thue-Siegel-Roth Theorem and Thue's Theorem (without proof of course). Description of Thue's method by a concrete example.
Section 2. Solving Thue equations by Baker's method (use of linear forms in logarithms of algebraic numbers); a list of book references for further study is given. Thue equation and Siegel's Unit equation. Lower bounds for linear forms in logarithms and Siege's Unit equation.
Section 3. Families of Thue equations and finiteness of solutions to such families. Short description of the author's research project jointly with Claude Levesque, and a related conjecture.
Section 4. A guide for further references.
For the entire collection see [Zbl 1444.11004].
Reviewer: Nikos Tzanakis (Iraklion)Intervals of small measure containing an algebraic number of given height.https://zbmath.org/1456.111372021-04-16T16:22:00+00:00"Kalosha, Nikolaĭ Ivanovich"https://zbmath.org/authors/?q=ai:kalosha.nikolai-ivanovich"Korlyukova, Irina Aleksandrovna"https://zbmath.org/authors/?q=ai:korlyukova.irina-aleksandrovna"Guseva, Elena Vasil'evna"https://zbmath.org/authors/?q=ai:guseva.elena-vasilevnaSummary: Rational numbers are uniformly distributed, even though distances between rational neighbors in a Farey sequence can be quite different. This property doesn't hold for algebraic numbers. In [``On the distribution of real algebraic numbers of the second degree'' (Russian), Vest. NAN Belaruss., Ser. Fiz.-Mat. Nauk 2013, No. 3, 54--63 (2013); J. Théor. Nombres Bordx. 29, No. 1, 179--200 (2017; Zbl 1420.11126)], \textit{D. V. Koleda} found the distribution function for real algebraic numbers of an arbitrary degree under their natural ordering.
It can be proved that the quantity of real algebraic numbers \(\alpha\) of degree \(n\) and height \(H( \alpha ) \le Q\) asymptotically equals \(c_1(n)Q^{n+1}\). Recently it was proved that there exist intervals of length \(Q^{- \gamma }\), \(\gamma >1\), free of algebraic numbers \(\alpha\), \(H( \alpha ) \le Q\), however for \(0 \le \gamma <1\) there exist at least \(c_2(n)Q^{n+1- \gamma }\) algebraic numbers in such intervals.
In this paper we show that special intervals of length \(Q^{- \gamma }\) may contain algebraic numbers even for large values of \(\gamma \), however their quantity doesn't exceed \(c_3Q^{n+1- \gamma }\). An earlier result by \textit{F. Götze} and \textit{A. G. Gusakova} [Dokl. Nats. Akad. Nauk Belarusi 59, No. 4, 11--16 (2015; Zbl 1372.11073)] was proved only for the case \(\gamma = \frac{3}{2} \).An extension of the digital method based on \(b\)-adic integers.https://zbmath.org/1456.111272021-04-16T16:22:00+00:00"Hofer, Roswitha"https://zbmath.org/authors/?q=ai:hofer.roswitha"Pirsic, Ísabel"https://zbmath.org/authors/?q=ai:pirsic.isabelSummary: We introduce a hybridization of digital sequences with uniformly distributed sequences in the domain of \(b\)-adic integers, \(\mathbb Z_b\), \(b\in\mathbb N \backslash \{1\}\), by using such sequences as input for generating matrices. The generating matrices are then naturally required to have finite row-lengths. We exhibit some relations of the `classical' digital method to our extended version, and also give several examples of new constructions with their respective quality assessments in terms of \(t\), \(\mathbf T\) and discrepancy.Some explicit computations in Arakelov geometry of abelian varieties.https://zbmath.org/1456.111062021-04-16T16:22:00+00:00"Gaudron, Éric"https://zbmath.org/authors/?q=ai:gaudron.ericSummary: Given a polarized complex abelian variety \((\mathsf{A}, \mathsf{L})\), a Gromov lemma makes a comparison between the sup and \(L^2\) norms of a global section of \(\mathsf{L}\). We give here an explicit bound which depends on the dimension, degree and injectivity diameter of \((\mathsf{A}, \mathsf{L})\). It rests on a more general estimate for the jet of a global section of \(\mathsf{L}\). As an application we deduce some estimates of the maximal slope of the tangent and cotangent spaces of a polarized abelian variety defined over a number field. These results are effective versions of previous works by Masser and Wüstholz on one hand and
\textit{J. B. Bost} [Duke Math. J. 82, No. 1, 21--70 (1996; Zbl 0867.14010)] on the other. They also improve some similar statements established by \textit{P. Graftieaux} in [Duke Math. J. 106, No. 1, 81--121 (2001; Zbl 1064.14045)].Generalized simultaneous approximation to \(m\) linearly dependent reals.https://zbmath.org/1456.111252021-04-16T16:22:00+00:00"Summerer, Leonhard"https://zbmath.org/authors/?q=ai:summerer.leonhardSummary: In order to analyse the simultaneous approximation properties of \(m\) reals, the parametric geometry of numbers studies the joint behaviour of the successive minima functions with respect to a one-parameter family of convex bodies and a lattice defined in terms of the \(m\) given reals. For simultaneous approximation in the sense of Dirichlet, the linear independence over \(\mathbb{Q}\) of these reals together with 1 is equivalent to a certain nice intersection property that any two consecutive minima functions enjoy. This paper focusses on a slightly generalized version of simultaneous approximation where this equivalence is no longer in place and investigates conditions for that intersection property in the case of linearly dependent irrationals.Revisiting approximate polynomial common divisor problem and noisy multipolynomial reconstruction.https://zbmath.org/1456.941192021-04-16T16:22:00+00:00"Xu, Jun"https://zbmath.org/authors/?q=ai:xu.jun"Sarkar, Santanu"https://zbmath.org/authors/?q=ai:sarkar.santanu"Hu, Lei"https://zbmath.org/authors/?q=ai:hu.leiSummary: In this paper, we present a polynomial lattice method to solve the Approximate Polynomial Common Divisor problem. This problem is the polynomial version of the well known approximate integer common divisor problem introduced by \textit{N. Howgrave-Graham} [Lect. Notes Comput. Sci. 2146, 51--66 (2001; Zbl 1006.94528)]. Our idea can be applied directly to solve the Noisy Multipolynomial Reconstruction problem in the field of error-correcting codes. Compared to the method proposed by \textit{C. Devet}, \textit{I. Goldberg} and \textit{N. Heninger} [``Optimally robust private information retrieval'', in: Proceedings of the 21st USENIX conference on security symposium, Security 2012, Bellevue, WA, USA, 2012. Berkeley, CA: USENIX Association. 15 p. (2012)], our approach is faster.
For the entire collection see [Zbl 1428.94012].Elliptic curves over finite fields with Fibonacci numbers of points.https://zbmath.org/1456.111162021-04-16T16:22:00+00:00"Bilu, Yuri"https://zbmath.org/authors/?q=ai:bilu.yuri-f"Gómez, Carlos A."https://zbmath.org/authors/?q=ai:gomez.carlos-alexis"Gómez, Jhonny C."https://zbmath.org/authors/?q=ai:gomez.jhonny-c"Luca, Florian"https://zbmath.org/authors/?q=ai:luca.florianThe paper studies and solves a curious problem: let \(E\)\, be an elliptic curve defined over a finite field \(\mathbb{F}_q\)\, with \(\sharp(E)=q+1-a\)\, and let us denote \(E_m(q,a);\,\, m\ge 1\)\, the number of points of \(E\)\, over \(\mathbb{F}_{q^m}\); let \(\{F_n\}_{n\ge 1}\)\, be the Fibonacci sequence. When the intersection of \(\{E_m(q,a)\}_{m\ge 1}\)\, and \(\{F_n\}_{n\ge 1}\)\, has two elements at least?, i.e. to find the solutions of \(\sharp E_{m_1} = F_{n_1}\),\, \(\sharp E_{m_2} = F_{n_2}\), etc.
Section 1 discusses the problem and formulates the obtained result (Theorem 1.1): the cardinal of the intersection is two for four couples \((q,a)\)\, and three in only a case (for \(q=a=2\)). The rest of the paper is devoted to the proof of that theorem.
Section 2 studies the equation \(E_m(q,a)=F_n\)\, in terms of linear forms in logarithms and discusses how the problem can be reduced to three cases: (i) \(q\)\, small (\(q\le 10.000\)), (ii) \(n_1\)\, small and (iii) \(m_2\)\, small. This provides a list of the possible values for \(q\).
Section 3 gathers some necessary tools regarding linear forms in logarithms and continued fractions. Assuming proved that \(n_2\le 1.000\)\, Section 4 first studies the problem for \(q\le 10.000\)\, and, using the computational package \texttt{Mathematica} finds the five solutions of Theorem 1.1. Then, for \(q> 10.000\),\, also using Mathematica, proves that there is no solution.
The rest of the paper assumes \(n_2> 1.000\)\, and Section 5 to 10 deals with the three cases (i), (ii) and (iii) and, always with the computational help of Mathematica (the paper says that ``the total calculation time for the Mathematica software for this paper was 20 days on 25 parallel desktop computers'') finishes the proof of Theorem 1.1.
Reviewer: Juan Tena Ayuso (Valladolid)The finite subgroups of \(\mathrm{SL}(3,\overline{F})\).https://zbmath.org/1456.200562021-04-16T16:22:00+00:00"Flicker, Yuval Z."https://zbmath.org/authors/?q=ai:flicker.yuval-zThe \textit{special linear group} \(\mathrm{SL}(n,F)\) of degree \(n\) over a field \(F\) is the set of all \(n\times n\) matrices with determinant \(1\), with the group operations of ordinary matrix multiplication and matrix inversion. In the paper under review, the author gives a complete exposition of the finite subgroups of \(\mathrm{SL}(3,\bar{F})\), where \(\bar{F}\) is a separably closed field of characteristic not dividing the order of the finite subgroup. This completes the earlier work of \textit{H. F. Blichfeldt} [Math. Ann. 63, 552--572 (1907; JFM 38.0192.03)]. Several elementary examples are illustrated by the author in the exposition.
Reviewer: Mahadi Ddamulira (Kampala)On simultaneous rational approximation to a real number and its integral powers. II.https://zbmath.org/1456.111232021-04-16T16:22:00+00:00"Badziahin, Dzmitry"https://zbmath.org/authors/?q=ai:badziahin.dzmitry-a"Bugeaud, Yann"https://zbmath.org/authors/?q=ai:bugeaud.yannSummary: For a positive integer \(n\) and a real number \(\xi\), let \(\lambda_n(\xi)\) denote the supremum of the real numbers \(\lambda\) for which there are arbitrarily large positive integers \(q\) such that \(\|q\xi\|, \|q\xi^2\|, \dots, \|q\xi^n\|\) are all less than \(q^{-\lambda}\). Here, \(\| \cdot \|\) denotes the distance to the nearest integer. We establish new results on the Hausdorff dimension of the set of real numbers \(\xi\) such that \(\lambda_n(\xi)\) is equal (or greater than or equal) to a given value.
For Part I, see [the author, Ann. Inst. Fourier 60, No. 6, 2165--2182 (2010; Zbl 1229.11100)].On approximations of solutions of the equation \(P(z,\ln z) = 0\) by algebraic numbers.https://zbmath.org/1456.111282021-04-16T16:22:00+00:00"Galochkin, Alexander"https://zbmath.org/authors/?q=ai:galochkin.alexandr"Godunova, Anastasia"https://zbmath.org/authors/?q=ai:godunova.anastasiaLet \(d_1\) and \(d_2\) be positive integers. Let \(P(x,y)\in\mathbb Z[x,y]\) such that \(\deg_x =d_1\) and \(\deg_y =d_2\). Then, the authors prove that for every \(\varepsilon >0\) and \(r>0\), the inequality \[|P(\theta,\log\theta)| <\exp\Big(-(4+\varepsilon)d_1d_2\frac{\ln^2L(\theta)}{\ln\ln L(\theta)}\Big)\] has only finitely many solutions in algebraic \(\theta\) such that \(|\theta| <r\) and \(\deg\theta =o(\ln\ln L(\theta))\) as \(L(\theta)\to\infty\). Here, \(L(\theta)\) denotes the length of the algebraic number \(\theta\), this is the sum of the coefficients in absolute value of its minimal polynomial. The proof is in the spirit of Mahler.
Reviewer: Jaroslav Hančl (Ostrava)On the degeneracy of integral points and entire curves in the complement of nef effective divisors.https://zbmath.org/1456.111192021-04-16T16:22:00+00:00"Heier, Gordon"https://zbmath.org/authors/?q=ai:heier.gordon"Levin, Aaron"https://zbmath.org/authors/?q=ai:levin.aaronSummary: As a consequence of the divisorial case of our recently established generalization of Schmidt's subspace theorem, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of our result is the attainment of a strong degeneracy conclusion (arithmetic quasi-hyperbolicity) under weak positivity assumptions on the divisors. The proof hinges on applying our recent theorem with a well-situated ample divisor realizing a certain lexicographical minimax. We also explore the connections with earlier work by other authors and make a conjecture regarding bounds for the numbers of divisors necessary, including consideration of the question of arithmetic hyperbolicity. Under the standard correspondence between statements in Diophantine approximation and Nevanlinna theory, one obtains analogous degeneration statements for entire curves.The irrationality measure of \(\pi\) is at most 7.103205334137\dots.https://zbmath.org/1456.111292021-04-16T16:22:00+00:00"Zeilberger, Doron"https://zbmath.org/authors/?q=ai:zeilberger.doron"Zudilin, Wadim"https://zbmath.org/authors/?q=ai:zudilin.wadimThe main result of this paper is that the irrationality measure exponent of the number \(\pi\) is less than \(7.103205334138\). The proof uses complex analysis, is based on clever calculating of special integral and is in the spirit of Salikov.
Reviewer: Jaroslav Hančl (Ostrava)Equidistribution of expanding translates of curves and Diophantine approximation on matrices.https://zbmath.org/1456.220032021-04-16T16:22:00+00:00"Yang, Pengyu"https://zbmath.org/authors/?q=ai:yang.pengyuOne can begin with author's abstract:
``We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space \(G/\Gamma\) of a semisimple algebraic group \(G\). We define two families of algebraic subvarieties of the associated partial flag variety \(G/\Gamma\), which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of \(m\times n\) real matrices whose image is not contained in any subvariety coming from these two families, Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner's theorem on measure rigidity for unipotent flows, and linearization technique.''
It is noted that many problems in number theory can be recast in the language of homogeneous dynamics. A survey is devoted to this fact, to the main problems of this research, and to the motivation of the present investigations. Notions that are useful for proving the main statements are recalled and explained.
The main results and several auxiliary statements are proven with explanations. Applications of the main results and also connections between these results and known researches are noted.
Reviewer: Symon Serbenyuk (Kyïv)A note on trace of powers of algebraic numbers.https://zbmath.org/1456.111322021-04-16T16:22:00+00:00"Philippon, Patrice"https://zbmath.org/authors/?q=ai:philippon.patrice"Rath, Purusottam"https://zbmath.org/authors/?q=ai:rath.purusottamThe paper under review deals with the problem of detecting algebraic integers amongst algebraic numbers, by looking at the traces of their powers. These questions were studied by \textit{G. Pólya} [Rend. Circ. Mat. Palermo 40, 1--16 (1915; JFM 45.0655.02)] and \textit{B. de Smit} [J. Number Theory 45, No. 1, 112--116 (1993; Zbl 0782.11027)], who showed that if \(\alpha\) is an algebraic number and \(\mathrm{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}(\alpha^i) \in \mathbb{Z}\) for some finite, explicit sequence of integers \(i\) (depending on the degree of \(\alpha\)), then \(\alpha\) is an algebraic integer. As a corollary, one sees that \(\alpha\) is an algebraic integer as soon as \(\mathrm{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}(\alpha^i) \in \mathbb{Z}\) for all but finitely many \(i \in \mathbb{N}\).
This last assertion is generalised by Theorem 2 of the paper under review, which shows that an algebraic number \(\alpha\) is an algebraic integer as soon as, for any fixed algebraic number \(\lambda\), the trace \(\mathrm{Tr}_{\mathbb{Q}(\alpha, \lambda)/\mathbb{Q}}(\lambda \alpha^i)\) is integral and non-zero for all but finitely many \(i \in \mathbb{N}\). It is necessary that these traces are non-zero, as the examples \(\alpha = 1/\sqrt{2}\) or \(\alpha = (1 + \sqrt{5})/\sqrt{2}\) show.
The rest of the paper under review (see in particular Theorem 5 and Theorem 12) aims to study pairs of non-zero algebraic numbers \(\alpha, \lambda\) such that \(\mathrm{Tr}_{F/K}(\lambda \alpha^i) = 0\) for infinitely many \(i \in \mathbb{N}\), where \(K \subseteq F\) are two number fields such that \(\mathbb{Q}(\alpha,\lambda) \subseteq F\). We mention in particular that Corollary 6 shows that \(\mathrm{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}(\alpha^i) = 0\) for infinitely many \(i \in \mathbb{N}\) if and only if \(\alpha \not\in \mathbb{Q}(\alpha^{h})\), where \(h\) denotes the order of the torsion part of the Galois group of \(\alpha\) (that is, the Galois group of the splitting field of the minimal polynomial of \(\alpha\)). Moreover, Corollary 13 proves the analogous result with the extension \(\mathbb{Q} \subseteq \mathbb{Q}(\alpha)\) replaced by \(\mathbb{Q}(\zeta) \subseteq \mathbb{Q}(\alpha,\zeta)\), where \(\zeta\) is a primitive root of unity of order \(h\).
Perhaps surprisingly, the proofs of the paper under review use some deep results from Diophantine approximation, such as the theorem of Skolem, Mahler and Lech (see the work of \textit{G. Hansel} [Theor. Comput. Sci. 43, 91--98 (1986; Zbl 0605.10007)] for an elementary proof). Such a theorem does not have a literal analogue for function fields of positive characteristic (see however the work of \textit{H. Derksen} [Invent. Math. 168, No. 1, 175--224 (2007; Zbl 1205.11030)] for a possible analogue). The last section of the paper under review shows that also some of the aforementioned results do not have a literal analogue over function fields of positive characteristic.
Reviewer: Riccardo Pengo (Lyon)Algebraic results for the values \(\vartheta_3(m\tau)\) and \(\vartheta_3(n\tau)\) of the Jacobi theta-constant.https://zbmath.org/1456.111312021-04-16T16:22:00+00:00"Elsner, Carsten"https://zbmath.org/authors/?q=ai:elsner.carsten"Luca, Florian"https://zbmath.org/authors/?q=ai:luca.florian"Tachiya, Yohei"https://zbmath.org/authors/?q=ai:tachiya.yoheiSet \(\mathbb H= \{ \tau\in\mathbb C ; \Im(\tau)>0\}\). Let \(m\) and \(n\) be distinct integers and \(\tau\in\mathbb H\). Then the authors prove that at least two of numbers \(e^{\pi i\tau}\), \(1+2\sum_{j=1}^\infty e^{\pi in\tau j^2}\) and \(1+2\sum_{j=1}^\infty e^{\pi im\tau j^2}\) are algebraically independent over \(\mathbb Q\). In particular, the two numbers \(1+2\sum_{j=1}^\infty e^{\pi in\tau j^2}\) and \(1+2\sum_{j=1}^\infty e^{\pi im\tau j^2}\) are algebraically independent over \(\mathbb Q\) for any \(\tau\in\mathbb H\) such that \(e^{\pi i\tau}\) is an algebraic number.
Reviewer: Jaroslav Hančl (Ostrava)Counter-examples in parametric geometry of numbers.https://zbmath.org/1456.111242021-04-16T16:22:00+00:00"Rivard-Cooke, Martin"https://zbmath.org/authors/?q=ai:rivard-cooke.martin"Roy, Damien"https://zbmath.org/authors/?q=ai:roy.damienSummary: Thanks to recent advances in parametric geometry of numbers, we know that the spectrum of any set of \(m\) exponents of Diophantine approximation to points in \(\mathbb{R}^n\) (in a general abstract setting) is a compact connected subset of \(\mathbb{R}^m\). Moreover, this set is semialgebraic and closed under coordinatewise minimum for \(n\le 3\). In this paper, we give examples showing that for \(n\ge 4\) each of the latter properties may fail.Apéry type recurrence relations.https://zbmath.org/1456.110172021-04-16T16:22:00+00:00"Centurión Fajardo, Alicia María"https://zbmath.org/authors/?q=ai:centurion-fajardo.alicia-maria"Céspedes Trujillo, Nancy"https://zbmath.org/authors/?q=ai:cespedes-trujillo.nancy"Moreno Roque, Eduardo"https://zbmath.org/authors/?q=ai:moreno-roque.eduardoSummary: In the last decade, the study of recurrence relations has obtained great relevance. The generation of rational approximants has been favored with the inclusion of recurrence relations for the study of the irrationality of certain mathematical constants, in particular Apéry's constant. This article presents, based on the Zeilberger algorithm, two recurrence relations of the Apéry type. For the result, the denominators of the rational approximations to the Apéry constant were modified.Computation of the fundamental units of number rings using a generalized continued fraction.https://zbmath.org/1456.111262021-04-16T16:22:00+00:00"Bruno, A. D."https://zbmath.org/authors/?q=ai:bruno.alexander-dSummary: A global generalization of continued fraction is proposed. It is based on computer algebra and can be used to find the best Diophantine approximations. This generalization provides a basis for computing the fundamental units of algebraic rings and for finding all solutions of a class of Diophantine equations. Examples in dimensions two, three, and four are given.On a generalization of a theorem of Popov.https://zbmath.org/1456.111902021-04-16T16:22:00+00:00"Huang, Jing-Jing"https://zbmath.org/authors/?q=ai:huang.jingjing"Li, Huixi"https://zbmath.org/authors/?q=ai:li.huixiSummary: In this paper, we obtain sharp estimates for the number of lattice points under and near the dilation of a general parabola, the former generalizing an old result of \textit{V. N. Popov} [Math. Notes 18, 1007--1010 (1976; Zbl 0327.10049)]. We apply Vaaler's lemma [\textit{J. D. Vaaler}, Bull. Am. Math. Soc., New Ser. 12, 183--216 (1985; Zbl 0575.42003)] and the Erdős-Turán inequality to reduce the two underlying counting problems to mean values of a certain quadratic exponential sums, whose treatment is subject to classical analytic techniques.On Diophantine transference principles.https://zbmath.org/1456.111302021-04-16T16:22:00+00:00"Ghosh, Anish"https://zbmath.org/authors/?q=ai:ghosh.anish"Marnat, Antoine"https://zbmath.org/authors/?q=ai:marnat.antoineThe present paper deals with Diophantine approximations on manifolds. A special attention is given to certain types of manifolds. Theorems from [\textit{V. Beresnevich} and \textit{S. Velani}, Proc. Lond. Math. Soc. 101, No.3, 821--851 (2010; Zbl 1223.11091)] are extended.
The authors present ``an extension of the transference results of Beresnevich and Velani connecting homogeneous and inhomogeneous Diophantine approximation on manifolds and provide bounds for inhomogeneous Diophantine exponents of affine subspaces and their nondegenerate submanifolds''. In this paper, a theorem of Beresnevich and Velani is extended ``from extremal transfer to transfer for arbitrary exponents and as a consequence, obtain the first known bounds for the inhomogeneous Diophantine exponent of affine subspaces and their nondegenerate submanifolds''. More general exponents of inhomogeneous Diophantine approximations are defined.
The authors discuss some open problems and difficulties for topics of the present investigation.
Reviewer: Symon Serbenyuk (Kyïv)