Recent zbMATH articles in MSC 11K https://www.zbmath.org/atom/cc/11K 2022-06-24T15:10:38.853281Z Werkzeug On a question of Zannier https://www.zbmath.org/1485.11098 2022-06-24T15:10:38.853281Z "Katz, Nicholas M." https://www.zbmath.org/authors/?q=ai:katz.nicholas-m Let $$E: y^2=4x^3-g_2 x-g_3$$ be an elliptic curve defined over $$\mathbb{Z}\left[\frac{1}{6N}\right]$$ ($$N$$ a positive integer), consider the differentials $$\omega=\frac{dx}{y}$$ (first kind) and $$\eta=\frac{x\,dx}{y}$$ (second kind) and apply the Cartier operator $$C_p$$ ($$p$$ a prime not dividing $$6N$$) to get $$C_p\omega=\alpha_p \omega$$ and $$C_p \eta=\beta_p\omega$$. The paper deals with a modular interpretation of $$(\alpha_p,\beta_p)$$ as $$p$$ varies: using the fact that $$\alpha_p$$ (resp. $$\beta_p$$) modulo $$p$$ is the coefficient of $$x^{p-1}$$ (resp. $$x^{p-2}$$) in $$(4x^3-g_2 x-g_3)^{\frac{p-1}{2}}$$ and the relation between $$g_2,g_3$$ and the Eisenstein series $$E_4,E_6$$, the author proves that $$\alpha_p$$ (resp. $$\beta_p$$) is the reduction modulo $$p$$ of $$E_{p-1}$$ (resp. $$\frac{E_{p+1}}{12}$$). The paper includes some computations for the ratio $$\frac{\alpha_p}{\beta_p} \pmod{p}$$ (which, in the CM case for a good ordinary $$p$$, turns out to provide a diagonalizing basis for the action of the order $$\mathcal{O}$$ on differentials) and for its distribution, and a series of similar open problems for curves of higher genus. Reviewer: Andrea Bandini (Pisa) New dimension bounds for $$\alpha \beta$$ sets https://www.zbmath.org/1485.11109 2022-06-24T15:10:38.853281Z "Baker, Simon" https://www.zbmath.org/authors/?q=ai:baker.simon.1|baker.simon Let $$\mathbb{T}:=\mathbb{R}/\mathbb{Z}$$ denote the unit circle. Given $$\alpha, \beta \in \mathbb{R}\backslash \mathbb{Q}$$, a nonempty closed set $$E \subset \mathbb{T}$$ is called an $$\alpha\beta$$ \textit{set} if for all $$x\in E$$ either $$x+\alpha ~\text{mod}~1 \in E$$ or $$x+\beta ~ \text{mod}~1 \in E$$. A sequence $$(x_n)_{n\ge 0}$$ of points in $$\mathbb{T}$$ is called an $$\alpha\beta$$ \textit{orbit} if for all $$n\ge 0$$, either $$x_{n+1}-x_n = \alpha ~\text{mod} ~1$$ or $$x_{n+1}-x_n = \beta ~\text{mod} ~1$$. Given $$\tau \ge 2$$, we say that $$x\in \mathbb{R}\setminus \mathbb{Q}$$ is $$\tau-$$well approximate if there exist infinitely many $$(p,q)\in \mathbb{Z}\times \mathbb{N}$$ satisfying \begin{align*} \left|x-\dfrac{p}{q}\right|< \dfrac{1}{q^{\tau}}. \end{align*} The set of $$\tau$$-well approximable numbers is denoted by $$W(\tau)$$. If $$\tau(x)=\infty$$, then we say that $$x$$ is a \textit{Liouville number}. If $$\tau \in [2, \infty) \cup \{\infty\}$$, the set of all real numbers with exact order $$\tau$$ is denoted by $$E(\tau)$$. In the paper under review, the author proves the following result, which is the main result in the paper. Theorem 1. Let $$\tau_1, \tau_2 \ge 2$$ satisfy $$2 \tau_1 < \tau_2+2$$ and suppose that $$\alpha \in E(\tau_1)$$ and $$\beta \in W(\tau_2)$$. Then any $$\alpha\beta$$ orbit $$(x_n)_{n\ge 0}$$ satisfies \begin{align*} \overline{\dim_{B}}(\{x_n\}) \ge 1-\frac{2(\tau_1-1)}{\tau_2}, \end{align*} where $$\overline{\dim_{B}}(\{x_n\})$$ is the upper box dimension of the $$\alpha\beta$$ set $$\{x_n\}$$. Furthermore, as an immediate corollary to Theorem 1, the author proves show that if $$\alpha$$ is not a Liouville number and $$\beta$$ is a Liouville number, then $$\overline{\dim_{B}}(\{x_n\}) = 1$$. The proof of Theorem 1 follow from a clever combination of techniques in number theory, the dimension theory, and the theory of continued fractions. Reviewer: Mahadi Ddamulira (Kampala) A random von Neumann theorem for uniformly distributed sequences of partitions https://www.zbmath.org/1485.11115 2022-06-24T15:10:38.853281Z "Carbone, Ingrid" https://www.zbmath.org/authors/?q=ai:carbone.ingrid Summary: In this paper, we prove a theorem that confirms, under a supplementary condition, a conjecture concerning random permutations of sequences of partitions of the unit interval. Recurrence relations and Benford's law https://www.zbmath.org/1485.11116 2022-06-24T15:10:38.853281Z "Farris, Madeleine" https://www.zbmath.org/authors/?q=ai:farris.madeleine "Luntzlara, Noah" https://www.zbmath.org/authors/?q=ai:luntzlara.noah "Miller, Steven J." https://www.zbmath.org/authors/?q=ai:miller.steven-j "Shao, Lily" https://www.zbmath.org/authors/?q=ai:shao.lily "Wang, Mengxi" https://www.zbmath.org/authors/?q=ai:wang.mengxi The paper under review presents several aspects of Benford's law and its connection with recurrence sequences. The relation between Benford's law and recurrence sequences with \emph{constant} coefficients is now classical and is recalled in the paper. The main aim of the paper is to present some relation between Benford's law and recurrence sequences with \emph{non-constant} coefficients. A sequence $$(a_n)_n$$ of positive real numbers is said to be (strongly)-Benford in base $$b$$ or simply \emph{Benford in base} $$b$$ if the sequence $$(\log_b(a_n))_n$$ is uniformly distributed modulo 1. This implies that the set of the integers $$n$$ for which the most significant digit of $$n$$ in base $$b$$ is $$d$$ has the asymptotic density $$\log_b (1+1/)d$$, the usual Benford's property. For example, if $$\rho$$ is a real number such that $$\log_b \rho$$ is irrational, the sequence $$(a_n)_n = (\rho^n)_n$$ (which is a solution of the recurrence relation $$a_{n+1}= \rho a_{n}$$) is Benford in base $$b$$. In a similar way, the Fibonacci sequence $$(F_n)_n$$ (solution of $$F_{n+2} = F_{n+1} + F_n$$) is Benford in base 10, since $$\sqrt{5}$$ and $$10$$ are mutiplicatively independent over $$\mathbb{Q}$$. To a sequence $$(a_n)_n$$ satisfying the recurrence relation $a_{n+1}=f(n)a_n+g(n)a_{n-1}$ the authors associate the sequences $$(\lambda(n))_n$$ and $$(\mu(n))_n$$ defined by (they are well defined with the required conditions on $$f$$ and $$g$$) $f(n) = \lambda(n) + \mu(n) \; ; \; g(n) = -\lambda(n-1) \mu(n).$ Their main results reads Theorem 1.3. With notation (1) and (2) and assuming that $$g(n)/f(n)^2$$ tends to $$0$$, the sequence $$(a_n)_n$$ is Benford if and only if $$\left(\prod_{i=1}^n \mu(i)\right)_n$$ is Benford. Some applications are given. Finally, the authors discuss generalizations to higher order recurrence sequences as well as to multiplicative recurrence sequences. Reviewer: Jean-Marc Deshouillers (Bordeaux) On Proinov's lower bound for the diaphony https://www.zbmath.org/1485.11117 2022-06-24T15:10:38.853281Z "Kirk, Nathan" https://www.zbmath.org/authors/?q=ai:kirk.nathan Summary: In 1986, \textit{P. D. Projnov} published an explicit lower bound for the diaphony of finite and infinite sequences of points contained in the $$d$$-dimensional unit cube [C. R. Acad. Bulg. Sci. 39, No. 9, 31--34 (1986; Zbl 0616.10042)]. However, his widely cited paper does not contain the proof of this result but simply states that this will appear elsewhere. To the best of our knowledge, this proof was so far only available in a monograph of \textit{P. D. Projnov} written in Bulgarian [Quantitative theory of uniform distribution and integral approximation (Bulgarian). Plovdiv: University of Plovdiv, Bulgaria (2000)]. The first contribution of our paper is to give a self contained version of Proinov's proof in English. Along the way, we improve the explicit asymptotic constants implementing recent, and corrected results of [\textit{A. Hinrichs} and \textit{L. Markhasin}, J. Complexity 27, No. 2, 127--132 (2011; Zbl 1215.65007); \textit{A. Hinrichs} and \textit{G. Larcher}, J. Complexity 34, 68--77 (2016; Zbl 1414.11087)]. (The corrections are due to a note in [Hinrichs and Larcher, loc. cit.].) Finally, as a main result, we use the method of Proinov to derive an explicit lower bound for the dyadic diaphony of finite and infinite sequences in a similar fashion. A generalized Faulhaber inequality, improved bracketing covers, and applications to discrepancy https://www.zbmath.org/1485.11118 2022-06-24T15:10:38.853281Z "Gnewuch, Michael" https://www.zbmath.org/authors/?q=ai:gnewuch.michael "Pasing, Hendrik" https://www.zbmath.org/authors/?q=ai:pasing.hendrik "Weiß, Christian" https://www.zbmath.org/authors/?q=ai:weiss.christian-h|weiss.christian Let $$A \subseteq [0,1]^d$$ and $$\delta > 0$$. The bracketing number $$N_{[ \; ]}(A, \delta)$$ of $$A$$ is the smallest number of closed axis parallel boxes $$[x,y] = [x_1, y_1] \times \dots \times [x_d, y_d]$$ needed to cover $$A$$, satisfying $$\lambda_d([0,y]) - \lambda_d([0,x]) \le \delta$$, where $$[0,x] = [0,x_1] \times \dots \times [0,x_d]$$ and similarly for $$y$$. Here $$\lambda_d$$ denotes the $$d$$-dimensional Lebesgue measure. The authors prove that $N_{[ \; ]}([0,1]^d, \delta) \le \max(1.1^{d-101}, 1)\frac{d^d}{d!} (\delta^{-1} + 1)^d.$ This improves substantially on previous bounds obtained by the first author [J. Complexity 24, No. 2, 154--172 (2008; Zbl 1138.11031)] and the latter two authors [Publ. Inst. Math., Nouv. Sér. 107(121), 67--74 (2020; Zbl 1474.11137)]. The bound on bracketing numbers is subsequently applied to provide new bounds on the star-discrepancy of negatively dependent random point sets, as well as a weighted variant of star-discrepancy for such sets. Reviewer: Simon Kristensen (Aarhus) An analogue of Pillai's theorem for continued fraction normality and an application to subsequences https://www.zbmath.org/1485.11119 2022-06-24T15:10:38.853281Z "Nandakumar, Satyadev" https://www.zbmath.org/authors/?q=ai:nandakumar.satyadev "Pulari, Subin" https://www.zbmath.org/authors/?q=ai:pulari.subin "Vishnoi, Prateek" https://www.zbmath.org/authors/?q=ai:vishnoi.prateek "Viswanathan, Gopal" https://www.zbmath.org/authors/?q=ai:viswanathan.gopal The authors give the following description of the present research: In the study of normal numbers using the base-$$b$$ representation, $$b\ge 2$$, we know that a number is normal if and only if it is simply normal in all the bases $$b^n$$, $$n\ge 1$$. We prove the analogue of this result for continued fraction normality. In particular, we show that two notions of continued fraction normality, one where overlapping occurrences of finite patterns are counted as distinct occurrences, and another where only disjoint occurrences are counted as distinct, are identical. This equivalence involves an analogue of a theorem due to \textit{S. S. Pillai} [Proc. Indian Acad. Sci., Sect. A 11, 73--80 (1940; Zbl 0023.20500)] for base-$$b$$ expansions. The proof requires techniques which are fundamentally different, since the continued fraction expansion utilizes a countably infinite alphabet, leading to a non-compact space. Utilizing the equivalence of these two notions, we provide a new proof of Heersink and Vandehey's recent result that selection of subsequences along arithmetic progressions does not preserve continued fraction normality [\textit{B. Heersink} and \textit{J. Vandehey}, Arch. Math. 106, No. 4, 363--370 (2016; Zbl 1337.11047)].'' A brief survey of this paper is devoted to the notions of the normality of numbers defined in terms of continued fractions and the base-$$b$$ representation. Auxiliary notions and notations are described. The special attention is given to continued fraction normal numbers, to the left-shift transformation, and to the Gauss measure. Several additional statements are proven. All results are given with explanations including the consideration of techniques which are used in proofs. Reviewer: Symon Serbenyuk (Kyïv) Metric properties about Banach averages and super simply normal numbers https://www.zbmath.org/1485.11120 2022-06-24T15:10:38.853281Z "Chen, Haibo" https://www.zbmath.org/authors/?q=ai:chen.haibo.1|chen.haibo.2|chen.haibo|chen.haibo.3 "Wang, Yi" https://www.zbmath.org/authors/?q=ai:wang.yi.1|wang.yi.8|wang.yi.5|wang.yi.6|wang.yi.10|wang.yi.9|wang.yi.3|wang.yi.2|wang.yi.4|wang.yi.7 "Xiao, Yu" https://www.zbmath.org/authors/?q=ai:xiao.yu Summary: In this paper, the concepts of Banach averages of digits of numbers and super simply normal numbers are introduced. Besides, the Lebesgue measure and Hausdorff dimension of set of super simply normal numbers are investigated. Moreover, by constructing suitable $$\alpha$$-Moran sets and \textbf{$$r$$}-Moran sets, the Hausdorff dimensions of level sets related to Banach averages are determined as well. Metrical problems in Diophantine approximation https://www.zbmath.org/1485.11121 2022-06-24T15:10:38.853281Z "Bakhtawar, Ayreena" https://www.zbmath.org/authors/?q=ai:bakhtawar.ayreena This paper deals with author's thesis which is devoted to several new results concerning metrical description of the sets of Dirichlet nonimprovable numbers. In Chapter 1, the attention is given to the main notions and a short overview of classical results in Diophantine approximation and its metrical theory. Chapter 2 is devoted to some auxiliary results from metrical theory along with elementary results on continued fractions.'' Chapter 3 includes the Lebesgue measure result for the set of Dirichlet nonimprovable numbers established by \textit{D. Kleinbock} and \textit{N. Wadleigh} [Proc. Am. Math. Soc. 146, No. 5, 1833--1844 (2018; Zbl 1448.11126)]''. Chapters 4--6 contain the main new results and solve three problems in uniform Diophantine approximation''. Brief explanations of presented results are given in this paper. Reviewer: Symon Serbenyuk (Kyïv) A brief and understandable guide to pseudo-random number generators and specific models for security https://www.zbmath.org/1485.65007 2022-06-24T15:10:38.853281Z "Almaraz Luengo, Elena" https://www.zbmath.org/authors/?q=ai:almaraz-luengo.elena Summary: The generation of random sequences is the basis of simulation and can be used in many different areas such as Statistics, Computer Science, Systems Management and Control, Biology, Particle Physics, Cryptography or Cyber-Security, among others. It is crucial that the numbers generated were random or at least, behave as such. The fundamental statistical properties required for such sequences are randomness and independence and, from a cryptographic perspective, unpredictability. There is a variety of methods to generate these sequences. The main ones are physical and arithmetic methods. In this work, a detailed study of the main arithmetic methods is carried out. On the other hand, the necessity of secure sequence generation will be analyzed and new lines of ongoing research focusing applications in Internet of Things and new generator designs will be described.