Abstract: For m, d ∈ ℕ, a jittered sample of N = md points can be constructed by partitioning [0, 1]d into md axis-aligned equivolume boxes and placing one point independently and uniformly at random inside each box. We utilise a formula for the expected ℒ2−discrepancy of stratified samples stemming from general equivolume partitions of [0, 1]d which recently appeared, to derive a closed form expression for the expected ℒ2−discrepancy of a jittered point set for any m, d ∈ ℕ. As a second main result we derive a similar formula for the expected Hickernell ℒ2−discrepancy of a jittered point set which also takes all projections of the point set to lower dimensional faces of the unit cube into account. PubDate: Thu, 10 Aug 2023 00:00:00 GMT

Abstract: We prove an easy statement about inhomogeneous approximation for non-singular vectors in metric theory of Diophantine approximation. PubDate: Thu, 10 Aug 2023 00:00:00 GMT

Abstract: Two-dimensional distribution function g(x, y) defined in [0, 1]2 is called copula, if g(x, 1) = x and g(1,y)= y for every x, y. Similarly, s-dimensional copula is a distribution function g(x1,x2,...,xs) such that every k-dimensional face functiong(1,…,1,xi1,1,…,1,xi2,1,…,1,xik,1,…,1)g\left( {1, \ldots ,1,{x_{{i_1}}},1, \ldots ,1,{x_{{i_2}}},1, \ldots ,1,{x_{{i_k}}},1, \ldots ,1} \right)is equal to xi1 xi2 ...xik for some but fixed k. In this paper we summarize and extend all known parts of copulas.In this paper we use the following abbreviations:{x} — fractional part of x;{x} — x mod 1;[x] — integer part of x;u.d. — uniform distribution;d.f. — distribution function;a.d.f. — asymptotic distribution function;u.d.p. — uniform distribution preserving;step d.f. — step distribution function;a.e. — almost everywhere;#X — cardinality of the set X. PubDate: Thu, 10 Aug 2023 00:00:00 GMT

Abstract: We give a necessary and sufficient condition for equidistribution of continuous functions along monotone compact covers on locally compact spaces. We show the existence of equidistributed mappings along Bohr nets arising from group actions. Using almost periodic means, we give an analogue of Weyl’s equidistribution criterion for continuous functions with values in arbitrary topological groups. We prove van der Corput’s inequality on the lattice ℕm for vectors in Hilbert spaces, and use this inequality to extend Hlawka’s equidistribution theorem to functions on the lattice ℕm (m ≥ 1) with values in arbitrary topological groups. PubDate: Thu, 10 Aug 2023 00:00:00 GMT

Abstract: It is important in cryptographic applications that the “key” used should be generated from a random seed. Thus, if the Legendre symbol sequence generated by a polynomial (as proposed by Hoffstein and Lieman) is used, that is{ (f(1)p),(f(2)p),(f(3)p),⋯,(f(p)p) },\left\{ {\left( {{{f\left( 1 \right)} \over p}} \right),\left( {{{f\left( 2 \right)} \over p}} \right),\left( {{{f\left( 3 \right)} \over p}} \right), \cdots ,\left( {{{f\left( p \right)} \over p}} \right)} \right\},then it is important to choose the polynomial f “almost” at random. Goubin, Mauduit, and Sárközy presented some not very restrictive conditions on the polynomial f, but these conditions may not be satisfied if we choose a “truly” random polynomial. However, how can it be guaranteed that the pseudorandom measures of the sequence should be small for almost "random" polynomials' These semirandom polynomials will be constructed with as few modifications as necessary from a truly random polynomial. PubDate: Thu, 10 Aug 2023 00:00:00 GMT

Abstract: For a probability measure μ on [0, 1] without discrete component, the best possible order of approximation by a finite point set in terms of the star-discrepancy is &inline as has been proven relatively recently. However, if μ contains a discrete component no non-trivial lower bound holds in general because it is straightforward to construct examples without any approximation error in this case. This might explain, why the approximation of discrete measures on [0, 1] by finite point sets has so far not been completely covered in the existing literature. In this note, we close the gap by giving a complete description for discrete measures. Most importantly, we prove that for any discrete measures (not supported on one point only) the best possible order of approximation is for infinitely many N bounded from below by &inline for some constant 6 ≥ c> 2 which depends on the measure. This implies, that for a finitely supported discrete measure on [0, 1]d the known possible order of approximation &inline is indeed the optimal one. PubDate: Thu, 10 Aug 2023 00:00:00 GMT

Abstract: In this paper, we study an efficient algorithm for constructing point sets underlying quasi-Monte Carlo integration rules for weighted Korobov classes. The algorithm presented is a reduced fast component-by-component digit-by-digit (CBC-DBD) algorithm, which is useful for situations where the weights in the function space show a sufficiently fast decay. The advantage of the algorithm presented here is that the computational effort can be independent of the dimension of the integration problem to be treated if suitable assumptions on the integrand are met. By considering a reduced digit-by-digit construction, we allow an integration algorithm to be less precise with respect to the number of bits in those components of the problem that are considered less important. The new reduced CBC-DBD algorithm is designed to work for the construction of lattice point sets, and the corresponding integration rules (so-called lattice rules) can be used to treat functions in different kinds of function spaces. We show that the integration rules constructed by our algorithm satisfy error bounds of almost optimal convergence order. Furthermore, we give details on an efficient implementation such that we obtain a considerable speed-up of a previously known CBC-DBD algorithm that has been studied in the paper Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness by Ebert, Kritzer, Nuyens, and Osisiogu, published in the Journal of Complexity in 2021. This improvement is illustrated by numerical results. PubDate: Thu, 10 Aug 2023 00:00:00 GMT

Abstract: In 1895, Vahlen proved a theorem concerning a simultaneous approximation of a real number by its two consecutive convergent. In this paper, we will provide a sharper bound for the theorem. PubDate: Thu, 10 Aug 2023 00:00:00 GMT

Abstract: For polynomials f (x),g1(x),g2(x)over ℤ, we report several observations about the density of primes p for which f (x) is fully splitting at p and { g1(r)p }<{ g2(r)p }\left\{ {{{{g_1}\left( r \right)} \over p}} \right\} < \left\{ {{{{g_2}\left( r \right)} \over p}} \right\} for some root r of f (x) ≡ 0 mod p. PubDate: Thu, 10 Aug 2023 00:00:00 GMT

Abstract: For sequences sufficiently close to (a log n), with an arbitrary real constant a, this note describes the precise asymptotics of the associated empirical distributions modulo one, with respect to the Kantorovich metric as well as a discrepancy-style metric. In particular, the note demonstrates how these asymptotics depend on a in a delicate, discontinuous way. The results strengthen and complement known facts in the literature. PubDate: Mon, 12 Dec 2022 00:00:00 GMT

Abstract: We examine the uniform distribution theory of H. Weyl when there is a periodic perturbation present. As opposed to the classical setting, in this case the conditions for (mod 1) density and (mod 1) uniform distribution turn out to be different. PubDate: Mon, 12 Dec 2022 00:00:00 GMT

Abstract: In this paper, we consider the hyperelliptic analogue of the Frobenius endomorphism generator and show that it produces sequences with large linear complexity on the Jacobian of genus 2 curves. PubDate: Mon, 12 Dec 2022 00:00:00 GMT

Abstract: Let (an)n=0∞ be a second-order linear recurrence sequence with constant coefficient. We study the limit points and asymptotic distribution of the sequence of consecutive ratios an+1/an. PubDate: Mon, 12 Dec 2022 00:00:00 GMT

Abstract: In this paper we study the distribution of leading digits of imaginary parts of Riemann zeta zeros in the b-adic expansion. PubDate: Mon, 12 Dec 2022 00:00:00 GMT

Abstract: Let K be a number field which is multiquadratic or quartic cyclic. We prove several results about the Kummer extensions of K, namely concerning the intersection between the Kummer extensions and the cyclotomic extensions of K. For G a finitely generated subgroup of K×, we consider the cyclotomic-Kummer extensions K(ζnt,Gn)/K(ζnt)K\left( {{\zeta _{nt}},\root n \of G } \right)/K\left( {{\zeta _{nt}}} \right) for all positive integers n and t, and we describe an explicit finite procedure to compute at once the degree of all these extensions. PubDate: Mon, 12 Dec 2022 00:00:00 GMT

Abstract: We examine four different notions of variation for real-valued functions defined on the compact ring of integers of a non-Archimedean local field, with an emphasis on regularity properties of functions with finite variation, and on establishing non-Archimedean Koksma inequalities. The first version of variation is due to Taibleson, the second due to Beer, and the remaining two are new. Taibleson variation is the simplest of these, but it is a coarse measure of irregularity and it does not admit a Koksma inequality. Beer variation can be used to prove a Koksma inequality, but it is order-dependent and not translation invariant. We define a new version of variation which may be interpreted as the graph-theoretic variation when a function is naturally extended to a certain subtree of the Berkovich affine line. This variation is order-free and translation invariant, and it admits a Koksma inequality which, for a certain natural family of examples, is always sharper than Beer’s. Finally, we define a Fourier-analytic variation and a corresponding Koksma inequality which is sometimes sharper than the Berkovich-analytic inequality. PubDate: Mon, 12 Dec 2022 00:00:00 GMT

Abstract: The Minkowski question-mark function '(x) is a continuous monotonous function defined on [0, 1] interval. It is well known fact that the derivative of this function, if exists, can take only two values: 0 and +∞.It isalso known that the value of the derivative ' (x)atthe point x =[0; a1,a2,...,at,...] is connected with the limit behaviour of the arithmetic mean (a1 +a2 +···+at)/t. Particularly, N. Moshchevitin and A. Dushistova showed that ifa1+a2+⋯+at<κ1,{a_1} + {a_2} + \cdots + {a_t} < {\kappa _1},where κ1=2log(1+52)/log2=1.3884…{\kappa _1} = 2\log \left( {{{1 + \sqrt 5 } \over 2}} \right)/\log 2 = 1.3884 \ldots, then '′(x)=+∞.They also proved that the constant κ1 is non-improvable. We consider a dual problem: how small can be the quantity a1 + a2 + ··· + at − κ1t if we know that ' (x) = 0' We obtain the non-improvable estimates of this quantity. PubDate: Mon, 12 Dec 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: 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