Abstract: The reduction modulo p of a family of lacunary integer polynomials, associated with the dynamical zeta function ζβ(z)of the β-shift, for β> 1 close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in 𝔽p and their factorizations, via Kronecker’s Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in 𝔽p is obtained in a few cases. In general it is expected since it must exist according to the Langlands program. PubDate: Tue, 31 May 2022 00:00:00 GMT

Abstract: Fix a positive integer N ≥ 2. For a real number x ∈ [0, 1] and a digit i ∈ {0, 1,..., N − 1}, let Πi(x, n) denote the frequency of the digit i among the first nN-adic digits of x. It is well-known that for a typical (in the sense of Baire) x ∈ [0, 1], the sequence of digit frequencies diverges as n →∞. In this paper we show that for any regular linear transformation T there exists a residual set of points x ∈ [0,1] such that the T -averaged version of the sequence (Πi(x, n))n also diverges significantly. PubDate: Tue, 31 May 2022 00:00:00 GMT

Abstract: In this paper we study the density in the real line of oscillating sequences of the form(g(k)⋅F(kα))k∈ℕ,{\left( {g\left( k \right) \cdot F\left( {k\alpha } \right)} \right)_{k \in \mathbb{N}}},where g is a positive increasing function and F a real continuous 1-periodic function. This extends work by Berend, Boshernitzan and Kolesnik [Distribution Modulo 1 of Some Oscillating Sequences I-III] who established differential properties on the function F ensuring that the oscillating sequence is dense modulo 1.More precisely, when F has finitely many roots in [0, 1), we provide necessary and also sufficient conditions for the oscillating sequence under consideration to be dense in ℝ. All the results are stated in terms of the Diophantine properties of α, with the help of the theory of continued fractions. PubDate: Tue, 31 May 2022 00:00:00 GMT

Abstract: Defined by Borel, a real number is normal to an integer base b ≥ 2 if in its base-b expansion every block of digits occurs with the same limiting frequency as every other block of the same length. We consider the problem of insertion in constructed base-b normal expansions to obtain normality to base (b + 1). PubDate: Tue, 31 May 2022 00:00:00 GMT

Abstract: Let s(n) be the number of nonzero bits in the binary digital expansion of the integer n. We study, for fixed k, ℓ, m, the Diophantine systems(ab)= k, s(a)= ℓ, and s(b)= min odd integer variables a, b.When k =2 or k = 3, we establish a bound on ab in terms of ℓ and m. While such a bound does not exist in the case of k =4, we give an upper bound for min{a, b} in terms of ℓ and m. PubDate: Tue, 31 May 2022 00:00:00 GMT

Abstract: In this paper, we discuss some properties of completely irrational subspaces. We prove that there exist completely irrational subspaces that are badly approximable and, moreover, sets of such subspaces are winning in different senses. We get some bounds for Diophantine exponents of vectors that lie in badly approximable subspaces that are completely irrational; in particular, for any vector ξ from two-dimensional badly approximable completely irrational subspace of ℝd one has ω⌢(ξ)≤5-12\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \omega } \left( \xi \right) \le {{\sqrt {5 - 1} } \over 2}. Besides that, some statements about the dimension of subspaces generated by best approximations to completely irrational subspace easily follow from properties that we discuss. PubDate: Tue, 31 May 2022 00:00:00 GMT

Abstract: In this note we give an overview of the currently known best lower and upper bounds on the size of a subset of ℤnm avoiding k-term arithmetic progression. We will focus on the case when the length of the forbidden progression is 3. We also formulate some open questions. PubDate: Tue, 31 May 2022 00:00:00 GMT

Abstract: Let K be a number field, and let ℓ be a prime number. Fix some elements α1,...,αr of K× which generate a subgroup of K× of rank r. Let n1,...,nr, m be positive integers with m ⩾ ni for every i. We show that there exist computable parametric formulas (involving only a finite case distinction) to express the degree of the Kummer extension K(ζℓm, α1ℓn1,…,αrℓnr\root {{\ell ^{{n_1}}}} \of {{\alpha _1}} , \ldots ,\root {{\ell ^{{n_r}}}} \of {{\alpha _r}}) over K(ζℓm) for all n1,..., nr, m. This is achieved with a new method with respect to a previous work, namely we determine explicit formulas for the divisibility parameters which come into play. PubDate: Wed, 02 Feb 2022 00:00:00 GMT

Abstract: In 1909 Borel defined normality as a notion of randomness of the digits of the representation of a real number over certain base (fractional expansion). If we think of the representation of a number over a base as an infinite sequence of symbols from a finite alphabet A, we can define normality directly for words of symbols of A: A word x is normal to the alphabet A if every finite block of symbols from A appears with the same asymptotic frequency in x as every other block of the same length. Many examples of normal words have been found since its definition, being Champernowne in 1933 the first to show an explicit and simple instance. Moreover, it has been characterized how we can select subsequences of a normal word x preserving its normality, always leaving the alphabet A fixed. In this work we consider the dual problem which consists of inserting symbols in infinitely many positions of a given word, in such a way that normality is preserved. Specifically, given a symbol b that is not present in the original alphabet A and given a word x that is normal to the alphabet A we solve how to insert the symbol b in infinitely many positions of the word x such that the resulting word is normal to the expanded alphabet A ∪{b}. PubDate: Wed, 02 Feb 2022 00:00:00 GMT

Abstract: We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and third-named authors for imaginary quadratic number fields [On the distribution of αp modulo one in imaginary quadratic number fields with class number one, J. Théor. Nombres Bordx. 32 (2020), no. 3, 719–760]) and by the first- and second-named authors for real quadratic number fields [Diophantine approximation with prime restriction in real quadratic number fields, Math. Z. (2021)] to 7/22. This generalizes a result of Harman [Diophantine approximation with Gaussian primes, Q. J. Math. 70 (2019), no. 4, 1505–1519] who obtained the same exponent 7/22 for ℚ (i) by extending his method which gave this exponent for ℚ [On the distribution of αp modulo one. II, Proc. London Math. Soc. 72, (1996), no. 3, 241–260]. Our proof is based on an extension of Harman’s sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in the above-mentioned work by the first- and second-named authors, for which we use analytic properties of Hecke L-functions with Größencharacters. PubDate: Wed, 02 Feb 2022 00:00:00 GMT

Abstract: Let q be a positive integer and 𝒮={x0,x1,⋯,xT−1}⊆ℤq={0,1,…,q−1}{\scr S} = \{{x_0},{x_1}, \cdots ,{x_{T - 1}}\}\subseteq {{\rm{\mathbb Z}}_q} = \{0,1, \ldots ,q - 1\} with 0≤x0<x1<⋯<xT−1≤q−1.0 \le {x_0} < {x_1} <\cdots< {x_{T - 1}} \le q - 1.. We derive from S three (finite) sequences: (1) For an integer M ≥ 2let (sn)be the M-ary sequence defined by sn ≡ xn+1 − xn mod M, n =0, 1,...,T − 2.(2) For an integer m ≥ 2let (tn) be the binary sequence defined by sn≡xn+1−xn mod M,n=0,1,⋯,T−2.\matrix{{{s_n} \equiv {x_{n + 1}} - {x_n}\,\bmod \,M,} & {n = 0,1, \cdots ,T - 2.}\cr} n =0, 1,...,T − 2.(3) Let (un) be the characteristic sequence of S, tn={1if 1≤xn+1−xn≤m−1,0,otherwise,n=0,1,…,T−2.\matrix{{{t_n} = \left\{{\matrix{1 \hfill & {{\rm{if}}\,1 \le {x_{n + 1}} - {x_n} \le m - 1,} \hfill\cr{0,} \hfill & {{\rm{otherwise}},} \hfill\cr}} \right.} & {n = 0,1, \ldots ,T - 2.}\cr} n =0, 1,...,q − 1.We st... PubDate: Wed, 02 Feb 2022 00:00:00 GMT

Abstract: K. Mahler’s conjecture: There exists no ξ ∈ ℝ+ such that the fractional parts {ξ(3/2)n} satisfy 0 ≤ {ξ(3/2)n} < 1/2 for all n = 0, 1, 2,... Such a ξ, if exists, is called a Mahler’s Z-number. In this paper we prove that if ξ is a Z-number, then the sequence xn = {ξ(3/2)n}, n =1, 2,... has asymptotic distribution function c0(x), where c0(x)=1 for x ∈ (0, 1]. PubDate: Wed, 02 Feb 2022 00:00:00 GMT

Abstract: In a family of Sn-fields (n ≤ 5), we show that except for a density zero set, the lower and upper bounds of the Euler-Kronecker constants are −(n − 1) log log dK+ O(log log log dK) and loglog dK + O(log log log dK), resp., where dK is the absolute value of the discriminant of a number field K. PubDate: Sat, 30 Oct 2021 00:00:00 GMT

Abstract: In this paper, we classify all solutions with cyclic and semi-cyclic semigroup supports of the functional equations arising from multiplication of quantum integers with fields of coefficients of characteristic zero. This also solves completely the classification problem proposed by Melvyn Nathanson and Yang Wang concerning the solutions, with semigroup supports which are not prime subsemigroups of ℕ, to these functional equations for the case of rational field of coefficients. As a consequence, we obtain some results for other problems raised by Nathanson concerning maximal solutions and extension of supports of solutions to these functional equations in the case where the semigroup supports are not prime subsemigroups of ℕ. PubDate: Sat, 30 Oct 2021 00:00:00 GMT

Abstract: The higher-dimensional generalization of the weighted q-adic sum-of-digits functions sq,γ(n), n =0, 1, 2,..., covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., d-dimensional van der Corput-Halton or d-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted q-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function g(x)= x implies the uniform distribution modulo one of the weighted q-adic sum-of-digits function sq,γ (n), n = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences h1sq, γ (n)+h2sq,γ (n +1), where h1 and h2 are integers such that h1 + h2 ≠ 0 and that the akin two-dimensional sequence sq,γ (n), sq,γ (n +1) cannot be uniformly distributed modulo one if q ≥ 3. The properties of the two-dimensional sequence sq,γ (n),sq,γ (n +1), n =0, 1, 2,..., will be instrumental in the proofs of the final section, where we show how the growth properties of the sequence of weights influence the distribution of values of the weighted sum-of-digits function which in turn imply a new property of the van der Corput sequence. PubDate: Sat, 30 Oct 2021 00:00:00 GMT

Abstract: In the present paper the author uses the function system Γℬsconstructed in Cantor bases to show upper bounds of the extreme and star discrepancy of an arbitrary net in the terms of the trigonometric sum of this net with respect to the functions of this system. The obtained estimations are inequalities of the type of Erdős-Turán-Koksma. These inequalities are very suitable for studying of nets constructed in the same Cantor system. PubDate: Sat, 30 Oct 2021 00:00:00 GMT

Abstract: We provide an algorithm to approximate a finitely supported discrete measure μ by a measure νN corresponding to a set of N points so that the total variation between μ and νN has an upper bound. As a consequence if μ is a (finite or infinitely supported) discrete probability measure on [0, 1]d with a sufficient decay rate on the weights of each point, then μ can be approximated by νN with total variation, and hence star-discrepancy, bounded above by (log N)N−1. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure μ, there exist finite sets whose star-discrepancy with respect to μ is at most (log N)d−12N−1{\left( {\log \,N} \right)^{d - {1 \over 2}}}{N^{ - 1}}. Moreover, we close a gap in the literature for discrepancy in the case d =1 showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure. PubDate: Sat, 30 Oct 2021 00:00:00 GMT

Abstract: Given a countably infinite group G acting on some space X, an increasing family of finite subsets Gn, x∈ X and a function f over X we consider the sums Sn(f, x) = ∑g∈Gnf(gx). The asymptotic behaviour of Sn(f, x) is a delicate problem that was studied under various settings. In the following paper we study this problem when G is a specific lattice in SL (2, ℤ ) acting on the projective line and Gn are chosen using the word metric. The asymptotic distribution is calculated and shown to be tightly connected to Minkowski’s question mark function. We proceed to show that the limit distribution is stationary with respect to a random walk on G defined by a specific measure µ. We further prove a stronger result stating that the asymptotic distribution is the limit point for any probability measure over X pushed forward by the convolution power µ∗n. PubDate: Fri, 25 Dec 2020 00:00:00 GMT