Recent zbMATH articles in MSC 28Ahttps://zbmath.org/atom/cc/28A2021-02-12T15:23:00+00:00WerkzeugSpectral asymptotics of Laplacians related to one-dimensional graph-directed self-similar measures with overlaps.https://zbmath.org/1452.351252021-02-12T15:23:00+00:00"Ngai, Sze-Man"https://zbmath.org/authors/?q=ai:ngai.sze-man"Xie, Yuanyuan"https://zbmath.org/authors/?q=ai:xie.yuanyuanSummary: For the class of graph-directed self-similar measures on \(\mathbb{R} \), which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the spectral dimension of the Laplacians defined by these measures. For the class of finitely ramified graph-directed self-similar sets, the spectral dimension of the associated Laplace operators has been obtained by \textit{B. M. Hambly} and \textit{S. O. G. Nyberg} [Proc. Edinb. Math. Soc., II. Ser. 46, No. 1, 1--34 (2003; Zbl 1038.35046)]. The main novelty of our results is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition.Approximation orders of real numbers by \(\beta\)-expansions.https://zbmath.org/1452.110972021-02-12T15:23:00+00:00"Fang, Lulu"https://zbmath.org/authors/?q=ai:fang.lulu"Wu, Min"https://zbmath.org/authors/?q=ai:wu.min.1|wu.min.2|wu.min"Li, Bing"https://zbmath.org/authors/?q=ai:li.bing.1The authors note the following investigations:
``We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their \(\beta\)-expansions with the exponential order \(\beta^{-n}\). Moreover, the Hausdorff dimensions of sets of the real numbers which are approximated by all other orders, are determined. These results are also applied to investigate the orbits of real numbers under \(\beta\)-transformation, the shrinking target type problem, the Diophantine approximation and the run-length function of \(\beta\)-expansions.''
In this paper, a survey is devoted to known results related with \(\beta\)-expansions. Basic definitions and properties for \(\beta\)-expansions are given. The separate attention is given to \(n\)-th cylinders defined in terms of \(\beta\)-expansions.
One can note the following main result of this research.
Let \(\beta>1\) be a fixed number and \(\lambda\) be the Lebesgue measure on \([0, 1]\),
\[
[0,1]\ni x=\sum^{\infty} _{k=1}{\frac{\varepsilon_k(x)}{\beta^k}}
\]
and
\[
\omega_n(x)=\sum^{n} _{k=1}{\frac{\varepsilon_k(x)}{\beta^k}}.
\]
Theorem. Let \(\beta>1\) be a real number. Then for \(\lambda\)-almost all \(x\in [0, 1)\),
\[
\lim_{n\to\infty}{\frac{1}{n}\log_{\beta}{(x-\omega_n(x))}}=-1.
\]
Several main results are related to the following set
\[
\left\{x\in[0,1): \liminf_{n\to\infty}{\frac{1}{\phi(n)}\log_{\beta}{(x-\omega_n(x))}}=-1\right\},
\]
where \(\phi\) is a positive function defined on the set of all positive integers.
All proofs are given with explanations.
Reviewer: Symon Serbenyuk (Kyïv)Isomorphism and bi-Lipschitz equivalence between the univoque sets.https://zbmath.org/1452.370282021-02-12T15:23:00+00:00"Jiang, Kan"https://zbmath.org/authors/?q=ai:jiang.kan"Xi, Lifeng"https://zbmath.org/authors/?q=ai:xi.lifeng"Xu, Shengnan"https://zbmath.org/authors/?q=ai:xu.shengnan"Yang, Jinjin"https://zbmath.org/authors/?q=ai:yang.jinjinIn this paper, the main attention is given to self-similar sets (with and without the open set condition) and to the bi-Lipschitz equivalence.
Some basic notions such as orbit sets, bi-Lipschitz equivalent metric spaces, quasi-Hölder equivalent metric spaces, quasi-Lipschitz equivalent metric spaces, Pisot numbers, and configuration sets are given. Several auxiliary lemmas are given. Also, the measure-theoretic isomorphism and the bi-Lipschitz equivalence between two sets are considered.
A uniform formula of the Hausdorff dimension of \(U_1\) is given and the existence of the unique measures of maximal entropy with respect to the lazy map for the closure of \(U_1\) and \(K\), where \(K\) is a self-similar set or an attractor for a certain iterated function system, is proven.
Finally, several auxiliary statements are proven and examples are given. The proofs of the main results are explained in detail. Final remarks are devoted to a brief description of open problems.
Reviewer: Symon Serbenyuk (Kyïv)De Finetti coherence and the product law for independent events.https://zbmath.org/1452.600032021-02-12T15:23:00+00:00"Mundici, Daniele"https://zbmath.org/authors/?q=ai:mundici.danieleSummary: In an earlier paper the present author proved that de Finetti coherence is preserved under taking products of coherent books on two finite sets of independent events. Conversely, in this note it is proved that product is the only coherence preserving operation on coherent books. Our proof shows that the traditional definition of stochastically independent classes of events actually follows from the combination of two more basic notions: boolean algebraic independence and de Finetti coherent betting system.The doubling metric and doubling measures.https://zbmath.org/1452.540142021-02-12T15:23:00+00:00"Flesch, János"https://zbmath.org/authors/?q=ai:flesch.janos"Predtetchinski, Arkadi"https://zbmath.org/authors/?q=ai:predtetchinski.arkadi"Suomala, Ville"https://zbmath.org/authors/?q=ai:suomala.villeLet \((X, \delta)\) be a metric space. For open \(V \subseteq X\) the authors define the set
\[
V_{*} = \bigcup \{B(x, 2r) \colon B(x, r) \subseteq V\},
\]
where \(B(x, r)\) is the open ball with the center \(x \in X\) and radius \(r > 0\). Write \(V_{*}^{0} = V\) and \(V_{*}^{n} = (V_{*}^{n-1})_{*}\) for any integer \(n \geqslant 1\). Let \(U \subseteq X\) be open. Then define the directed doubling distance \(d_{\to}(V, U)\) and the doubling distance \(d(U, V)\) as
\[
d_{\to}(V, U) = \inf \{n \colon U \subseteq V_{*}^{n}\} \quad \text{and} \quad d(U, V) = \max\{d_{\to}(U, V), d_{\to}(V, U)\}.
\]
Recall that a measure \(\mu\) on a metric space \((X, \delta)\) is said to be \(C\)-doubling for \(C \in [1, \infty)\) if
\[
0 < \mu (B(x, 2r)) \leqslant C \mu (B(x, r)) < \infty
\]
holds for every open ball \(B(x, r)\) and denote by \(D_C(X)\) the set of all \(C\)-doubling measures on \(X\).
The authors find several connections between the doubling distance and the doubling measures.
\medskip
Theorem. Let \((X, \delta)\) be a metric space, let \(U\) and \(V\) be open subsets of \(X\) with \(d_{\to}(U, V) < \infty\) and let \(\mu \in D_C(X)\) for some \(C \in [1, \infty)\). Then the inequality
\[
\mu(U) \geqslant C^{-3d_{\to}(U, V)} \mu (V)
\]
holds.
For Borel subsets \(U\), \(V\) of \(X\) the authors introduce the quantities
\[
m_{\to}(U, V) = \inf_{C \in [1, \infty)} \inf \{t \geqslant 0 \colon \mu(U) \geqslant C^{-t} \mu(V), \mu \in D_C(X)\}
\]
and \(m(U, V) = \max \{m_{\to}(U, V), m_{\to}(V, U)\}\).
In the next theorem, that is a main result of the paper, a subset of a metric space is called a simple open set if it can be written as a finite union of open balls.
Theorem. Let \((X, \delta)\) be a metric space that carries a doubling measure. If \(U\) and \(V\) are simple open sets, then \(d(U, V) \leqslant 4[6m(U, V) + 2]\) holds.
The paper contains also a subtle game-theoretic definition of directed doubling distance and some interesting facts related to Lipschitz functions with respect to such distances.
Reviewer: Aleksey A. Dovgoshey (Slovyansk)An extension of the Baire property.https://zbmath.org/1452.031002021-02-12T15:23:00+00:00"Caruvana, Christopher"https://zbmath.org/authors/?q=ai:caruvana.christopher"Kallman, Robert R."https://zbmath.org/authors/?q=ai:kallman.robert-rThe set \(\mathcal{M}(X)\) of all Borel probability measures on a Polish space \(X\) carries a natural topology, which makes it a Polish space,
see [\textit{K. R. Parthasarathy}, Probability measures on metric spaces. New York-London: Academic Press (1967; Zbl 0153.19101)]. For a subset \(A\) of \(X\), let \(\mathcal{N}(A)\) be the set of \(\mu\) in \(\mathcal{M}(X)\) with the property that \(\mu^*(A)=0\) -- the annihilator of \(A\). One says that \(A\) is residually null is \(\mathcal{N}(A)\) is co-meager in \(\mathcal{M}(X)\). The authors show that every meager set is residually null; this implies that sets with the Baire property are residually measurable, i.e., the set of \(\mu\) in \(\mathcal{M}(X)\) for the set in question is \(\mu\)-measurable is co-meager. In fact, among sets with the Baire property meagerness and residual nullness are equivalent, thus, e.g., in \textit{R. M. Solovay}'s model [Ann. Math. (2) 92, 1--56 (1970; Zbl 0207.00905)] the two notions are equivalent. Whether this equivalence is consistent with ZFC is stated as an open problem; it cannot be a theorem as, for example, Luzin sets are non-meager and even universally null.
In the main portion of the paper, the authors define and study the extended Baire property for sets in Polish spaces by relaxing meager to residually null: \(A\) has the EBP if the is an open set such that \(U\triangle A\) is residually null. Many properties known for sets with the Baire property also hold for sets with the extended Baire property.
Reviewer: K. P. Hart (Delft)Intermediate value property for the Assouad dimension of measures.https://zbmath.org/1452.280072021-02-12T15:23:00+00:00"Suomala, Ville"https://zbmath.org/authors/?q=ai:suomala.villeLet \(X\) be a metric space. For \(x\in X\) and \(r>0\) let \(B(x,r)\) denote the closed \(r\)-ball with center \(x\). For \(x\in X\) and \(0<r<R\) let \(N(x,R,r)\) denote the minimal number of closed \(r\)-balls which cover \(B(x,R)\). The Assouad dimension of \(X\) is \(\operatorname{dim}_AX=\inf\bigl\{a\ge0:\exists C>0\) \(\forall x\in X\) \(\forall r>0\) \(\forall R>r\) \(N(x,R,r)\le C\bigl(\frac Rr\bigr)^a\bigr\}\). If \(\mu\) is a Borel regular outer measure on \(X\), the Assouad dimension of \(\mu\) is
\(\operatorname{dim}_A\mu=\inf\bigl\{a\ge0: \exists C>0\) \(\forall x\in X\) \(\forall r>0\) \(\forall R>r\) \(\frac{\mu(B(x,R))}{\mu(B(x,r))}\le C\bigl(\frac Rr\bigr)^a\bigr\}\). Recently, \textit{K. E. Hare} et al. [Proc. Am. Math. Soc. 148, No. 9, 3881--3895 (2020; Zbl 1443.28004)] have shown that if \(X\) is a compact set of reals with \(0<\operatorname{dim}_AX<\infty\), then for every real \(s>\operatorname{dim}_AX\), \(X\) supports a measure with Assouad dimension \(s\). In the paper under review the author proves this theorem for all complete metric spaces \(X\).
Reviewer: Miroslav Repický (Košice)On dichotomy law for beta-dynamical system in parameter space.https://zbmath.org/1452.110982021-02-12T15:23:00+00:00"Lü, Fan"https://zbmath.org/authors/?q=ai:lu.fan"Wu, Jun"https://zbmath.org/authors/?q=ai:wu.junThe present paper deals with beta-transformations \(T_\beta\) for any \(\beta>1\). The main attention is given to the speed of convergence in
\[
\liminf_{n\to\infty}{|T^n _{\beta}{1}-0|}=0.
\]
The following set of parameters \(\beta>1\) for which the point 0 can be well approximated by the orbit of 1 under the beta-transformation with given speed, is considered:
\[
E(0,\varphi)=\{\beta>1: |T^n _{\beta}{1}-0|<\varphi(n) \text{ for infinitely many } n\in\mathbb N\},
\]
where \(\varphi: \mathbb N \to (0,1]\) is a positive function.
For the last-mentioned set, the dichotomy law for the Lebesgue measure is obtained, i.e., some conditions under which the Lebesgue measure of this set is zero or is full, are proven. In addition, the Lebesgue measure of the following set is investigated:
\[
\{\beta>1: |T^n _{\beta}{1}-0|<\beta^{-l_n} ~\text{for infinitely many}~ n\in\mathbb N\},
\]
where \((l_n)\) is a given sequence of non-negative real numbers, \(n\in\mathbb N\).
Some notions and known results about beta-expansions are given, several auxiliary statements are proven. Proofs are given with explanations. For a general target, difficulties of the present investigations are discussed.
Reviewer: Symon Serbenyuk (Kyïv)Symmetric itinerary sets: algorithms and nonlinear examples.https://zbmath.org/1452.370472021-02-12T15:23:00+00:00"Harding, Brendan"https://zbmath.org/authors/?q=ai:harding.brendanSummary: We describe how to approximate fractal transformations generated by a one-parameter family of dynamical systems \(W:[0,1]\rightarrow [0,1]\) constructed from a pair of monotone increasing diffeomorphisms \(W_{i}\) such that \(W_{i}^{-1}:[0,1]\rightarrow [0,1]\) for \(i=0,1\). An algorithm is provided for determining the unique parameter value such that the closure of the symbolic attractor \(\overline{\Omega}\) is symmetrical. Several examples are given, one in which the \(W_{i}\) are affine and two in which the \(W_{i}\) are nonlinear. Applications to digital imaging are also discussed.The proto-Lorenz system in its chaotic fractional and fractal structure.https://zbmath.org/1452.370862021-02-12T15:23:00+00:00"Doungmo Goufo, Emile Franc"https://zbmath.org/authors/?q=ai:doungmo-goufo.emile-francDimension drop for harmonic measure on Ahlfors regular boundaries.https://zbmath.org/1452.310062021-02-12T15:23:00+00:00"Azzam, Jonas"https://zbmath.org/authors/?q=ai:azzam.jonasThis article discusses the dimension of harmonic measure for a connected domain in \(\mathbb{R}^{d+1}\). The main result of the article establishes that for any domain \(\Omega\subset\mathbb{R}^{d+1}\), with uniformly non-flat Ahlfors \(s\)-regular boundary with \(s \geq d\), the dimension of its harmonic measure is strictly less than \(s\).
Reviewer: Marius Ghergu (Dublin)Orthogonal exponentials of self-affine measures on \(\mathbb{R}^n\).https://zbmath.org/1452.280062021-02-12T15:23:00+00:00"Su, Juan"https://zbmath.org/authors/?q=ai:su.juan"Chen, Ming-Liang"https://zbmath.org/authors/?q=ai:chen.ming-liangLet \(M = \text{diag}(\rho_1^{-1}, \rho_2^{-1}, \ldots, \rho_n^{-1})\) be an expanding \(n\times n\) matrix with real entries, \(B\) a non-singular \(n\times n\) matrix with integer entries, and \(\widetilde{D}\subset \mathbb{Z}^n\). Denote by \(D\) the finite integer digit set \(B\widetilde{D}\) and let \(\mu_{M,D}\) be the self-affine measure generated by \(M\) and \(D\). The authors prove that if
\[
\{x\in [0,1]^n : \sum\limits_{\widetilde{d}\in \widetilde{D}} \exp (2\pi i \langle \widetilde{d}, x \rangle)=0\} = \{\frac{1}{s}(s_1, s_2, \ldots, s_n)^\top : 0 \leq s_1, \dots, s_n \leq s-1\}\setminus\{0\},
\]
for some integer \(s\geq 2\), then \(L^2(\mu_{M,D})\) admits an infinite orthogonal set of exponential functions iff there exist \(\rho_j\), \(j\in \{1, \ldots, n\}\), such that \(|\rho_j| = (q_j/p_j)^{1/r_j}\), for some \(p_j, q_j, r_j \in \mathbb{N}\) with \(\gcd(p_j,q_j) = 1\) and \(\gcd(p_j, s) > 1\).
Reviewer: Peter Massopust (München)Representation of solutions of a second order delay differential equation.https://zbmath.org/1452.340702021-02-12T15:23:00+00:00"Qiu, Kee"https://zbmath.org/authors/?q=ai:qiu.kee"Wang, Jinrong"https://zbmath.org/authors/?q=ai:wang.jinrongThe authors study an inhomogeneous second-order delay differential equation on the fractal set \(R^{\alpha n}\) \((0 < \alpha \leq 1)\) by using the theory of local calculus as discussed in [\textit{X.-J. Yang}, Advanced local fractional calculus and its applications. New York, NY: World Science Publisher (2012)]. They introduce delay cosine and sine type matrix functions and provide their properties on the fractal set. Further, the authors give the solutions representation to second-order differential equations with pure delay and equation with two delays.
Reviewer: Krishnan Balachandran (Coimbatore)Ultrafilters and maximal linked systems: basic relations.https://zbmath.org/1452.280032021-02-12T15:23:00+00:00"Chentsov, Aleksandr Georgievich"https://zbmath.org/authors/?q=ai:chentsov.alexander-gUltrafilters and maximal linked systems with elements in the form of sets from the fixed \(\pi\)-system with `zero' and `unit' are investigated. Special attention is given to the description of the set of maximal linked systems that are not ultrafilters. Necessary and sufficient conditions for the existence of the above-mentioned systems and some topological properties for the set of all maximal linked systems of this type are obtained. Examples are given for which all maximal linked systems are ultrafilters; this corresponds to the realization of the supercompact ultrafilter space when the topology of Wallman type is used.
Reviewer: Daniele Puglisi (Catania)The Sierpiński gasket as the Martin boundary of a non-isotropic Markov chain.https://zbmath.org/1452.310182021-02-12T15:23:00+00:00"Kesseböhmer, Marc"https://zbmath.org/authors/?q=ai:kessebohmer.marc"Samuel, Tony"https://zbmath.org/authors/?q=ai:samuel.tony"Sender, Karenina"https://zbmath.org/authors/?q=ai:sender.kareninaThis article discusses the Sierpinski gasket as the Martin boundary of a non-isotropic Markov chain. More precisely, the authors construct a class of non-isotropic Markov chains dependent on a parameter \(p\in (0, 1/2)\) and show that the Martin boundary (which is shown to be homeomorphic to the Sierpinsky gasket) and the minimal Martin boundary are distinct and independent of the choice of \(p\). Further, a harmonic structure is described on this non-isotropic Markov chain.
Reviewer: Marius Ghergu (Dublin)On continuous images of self-similar sets.https://zbmath.org/1452.280042021-02-12T15:23:00+00:00"Li, Yuanyuan"https://zbmath.org/authors/?q=ai:li.yuanyuan"Fan, Jiaqi"https://zbmath.org/authors/?q=ai:fan.jiaqi"Gu, Jiangwen"https://zbmath.org/authors/?q=ai:gu.jiangwen"Zhao, Bing"https://zbmath.org/authors/?q=ai:zhao.bing"Jiang, Kan"https://zbmath.org/authors/?q=ai:jiang.kanLet \(f:\mathbb{R}^2\to \mathbb{R}\) be a continuous function and let \(K_1\) and \(K_2\) be self-similar sets on \(\mathbb{R}\). The authors address the question of what the exact form of the continuous image of such sets under \(f\) is. Specifically, they show that for two Moran sets \(E_1\) and \(E_2\), the image \(f(E_1,E_2)\) is either a closed interval or the finite union of closed intervals provided the first order partial derivatives of \(f\) satisfy certain conditions.
Reviewer: Peter Massopust (München)On the existence of the Hutchinson measure for generalized iterated function systems.https://zbmath.org/1452.280052021-02-12T15:23:00+00:00"Strobin, Filip"https://zbmath.org/authors/?q=ai:strobin.filipThe present paper deals with the theory of iterated function systems (IFSs) and the theory of generalized iterated function systems.
The notions of generalized IFSs, the Hutchinson measure, and the Markov operator are explained. It is noted that the class of GIFSs' attractors is essentially wider than the class of IFSs' attractors. Special attention is given to the notion of the code space and to a result which shows that ``the relationships between a GIFS and its code space are similar as in the classical IFS case''. In addition, several other auxiliary notions and known results are considered.
A full description of iterations of the generalized Markov operator is given. Several auxiliary statements are proven. Also, it is shown that ``a probabilistic GIFS consisting of generalized Matkowski contractions on a complete metric space generates the Hutchinson measure''.
The notion of a topologically contracting GIFS (TGIFS) is described. For TGIFSs on metric spaces, a version of the the main result of this paper is obtained. Finally, several open problems related to the present investigations are noted and discussed.
Reviewer: Symon Serbenyuk (Kyïv)The smallest semicopula-based universal integrals: remarks and improvements.https://zbmath.org/1452.280022021-02-12T15:23:00+00:00"Borzová, Jana"https://zbmath.org/authors/?q=ai:borzova.jana"Halčinová, Lenka"https://zbmath.org/authors/?q=ai:halcinova.lenka"Hutník, Ondrej"https://zbmath.org/authors/?q=ai:hutnik.ondrejSummary: We provide several improvements and corrections of results dealing with the smallest universal integral \(\mathbf{I}_S\) based on a semicopula \(S\). We deal with a transformation of the integral into another semicopula-based universal integral for continuous and commutative semicopulas, monotone convergence theorems with new proofs and Fréchet-type mappings' properties corrected. During this way we discover several fine properties of the integral functional \(\mathbf{I}_S\). We also discuss further convergences of measurable functions and the corresponding integrals in the light of the newest results from the literature answering several open questions stated in our previous articles of the series.Disintegration and Bayesian inversion via string diagrams.https://zbmath.org/1452.180082021-02-12T15:23:00+00:00"Cho, Kenta"https://zbmath.org/authors/?q=ai:cho.kenta"Jacobs, Bart"https://zbmath.org/authors/?q=ai:jacobs.bartDisintegration and Bayesian probability are standard tools in
probability theory and statistics (Partharasathy, Probability Measures
on Metric Spaces; Bogachev, Measure Theory 1+2); their application can
be quite demanding from a measure theoretic point of view. The
present paper interprets them through string diagrams, providing a
new visual wrapper to well known contents in a
unified way. The authors introduce
these diagrams, which support an attractive graphic representation of
the otherwise rather abstract scenarios of (conditional)
probabilities. Their algebraic treatment is expressed in terms of
types and channels in a symmetric monoidal category. It is shown how
most of the results using disintegration obtained in the area of
stochastic transition systems [\textit{P. Panangaden}, Labelled Markov transition systems. London: Imperial College Press (2009)]
are interpreted in terms of string diagrams. The paper misses,
however, the existing work on inverting stochastic relations, which is
essentially based on disintegration and Bayesian inversion. The
formalism is apparently supported by a graphic framework based on the
Python programming language [\textit{K. Cho} and \textit{B. Jacobs}, LIPIcs -- Leibniz Int. Proc. Inform. 72, Article 25, 8 p. (2017; Zbl 1433.68266)]; some expressive
samples are indicated. Discrete probabilities are covered as well,
and we find some pretty examples illuminating the descriptive power
of this formalism. Abstracting away the formalisms of measure theory
shows also that there are other categories for which, e.g.,
disintegration as outlined here is a sensible notion (mentioned are
the non-empty powerset monad and the category of commutative
\(C^*\)-algebras).
Reviewer: Ernst-Erich Doberkat (Dortmund)On the range of simple symmetric random walks on the line.https://zbmath.org/1452.600282021-02-12T15:23:00+00:00"Chen, Yuan-Hong"https://zbmath.org/authors/?q=ai:chen.yuanhong"Wu, Jun"https://zbmath.org/authors/?q=ai:wu.junIn this article, the authors study the random walk \((S_n(x))_{n \geq 0}\) defined by the dyadic expansion of the real number \(x \in [0,1]\). If \(x\) is chosen according to the Lebesgue measure on \([0,1]\), then \((S_n(x))_{n \geq 0}\) is a simple symmetric random walk. In that case, \textit{P. Révész} [Random walk in random and non-random environments. Singapore etc.: World Scientific (1990; Zbl 0733.60091)] showed that the range \(R_n(x) := \#\{S_j(x),0 \leq j \leq n \}\) of this random walk is almost surely of order \((n \log \log n)^{1/2}\) for \(n\) large enough. In this article, the authors compute the Hausdorff dimension of the set of points \(x \in [0,1]\) such that \(R_n(x) \sim c n^\gamma\) for \(c > 0\) and \(0 < \gamma \leq 1\). They proved the Hausdorff dimension to be \(1\) if \(\gamma < 1\), and that for all \(c \in (0,1)\):
\[
\dim_H(\{x\in[0,1]:R_n(x)\sim cn\})=-\left(\tfrac{1+c}{2} \log_2\left( \tfrac{1+c}{2}\right) + \tfrac{1-c}{2} \log_2 \left(\tfrac{1-c}{2}\right) \right).
\]
Reviewer: Bastien Mallein (Paris)Non-Diophantine arithmetics in mathematics, physics and psychology.https://zbmath.org/1452.000132021-02-12T15:23:00+00:00"Burgin, Mark"https://zbmath.org/authors/?q=ai:burgin.mark"Czachor, Marek"https://zbmath.org/authors/?q=ai:czachor.marekPublisher's description: For a long time, all thought there was only one geometry -- Euclidean geometry. Nevertheless, in the 19th century, many non-Euclidean geometries were discovered. It took almost two millennia to do this. This was the major mathematical discovery and advancement of the 19th century, which changed understanding of mathematics and the work of mathematicians providing innovative insights and tools for mathematical research and applications of mathematics.
A similar event happened in arithmetic in the 20th century. Even longer than with geometry, all thought there was only one conventional arithmetic of natural numbers -- the Diophantine arithmetic, in which \(2+2=4\) and \(1+1=2\). It is natural to call the conventional arithmetic by the name Diophantine arithmetic due to the important contributions to arithmetic by Diophantus. Nevertheless, in the 20th century, many non-Diophantine arithmetics were discovered, in some of which \(2+2=5\) or \(1+1=3\). It took more than two millennia to do this. This discovery has even more implications than the discovery of new geometries because all people use arithmetic.
This book provides a detailed exposition of the theory of non-Diophantine arithmetics and its various applications. Reading this book, the reader will see that on the one hand, non-Diophantine arithmetics continue the ancient tradition of operating with numbers while on the other hand, they introduce extremely original and innovative ideas.