Volume 15 (2020): Edition 1 (June 2020)

In 2009, Grant, Shallit, and Stoll [Acta Arith. 140 (2009), [345–368] constructed a large family of pseudorandom sequences, called generalized Rudin--Shapiro sequences, for which they established some results about the average of discrete correlation coefficients of order 2 in cases where the size of the alphabet is a prime number or a squarefree product of primes. We establish similar results for an even larger family of pseudorandom sequences, constructed via difference matrices, in the case of an alphabet of any size. The constructions generalize those from Grant et al. In the case where the size of the alphabet is squarefree and where there are at least two prime factors, we obtain an improvement in the error term by comparison with the result of Grant et al.

In the present paper the so-called (Vil_Bs; α; γ)-diaphony as a quantitative measure for the distribution of sequences and nets is considered. A class of two-dimensional nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu } of type of Zaremba-Halton constructed in a generalized B₂-adic system or Cantor system is introduced and the (Vil_B2; α; γ)-diaphony of these nets is studied. The influence of the vector α = (α₁, α₂) of exponential parameters to the exact order of the (Vil_B2; α; γ)-diaphony of the nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu } is shown. If α₁ = α₂, then the following holds: if 1 < α₂ < 2 the exact order is 𝒪 (logNN1-ε{{\sqrt {\log N} } \over {{N^{1 - \varepsilon }}}}) for some ε > 0, if α₂ = 2 the exact order is 𝒪 (logNN{{\sqrt {\log N} } \over N}) and if α₂ > 2 the exact order is 𝒪 (logNN1+ε{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}) for some ε > 0. If α₁ > α₂, then the following holds: if 1 < α₂ < 2 the exact order is 𝒪 (logNN1-ε{{\sqrt {\log N} } \over {{N^{1 - \varepsilon }}}}) for some ε > 0, if α₂ = 2 the exact order is 𝒪 (1N{1 \over N}) and if α₂ > 2 the exact order is 𝒪 (logNN1+ε{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}) for some ε > 0. Here N = B_ν, where B_ν denotes the number of the points of the nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu }.

We exhibit a class of Littlewood polynomials that are not L^α-flat for any α ≥ 0. Indeed, it is shown that the sequence of Littlewood polynomials is not L^α-flat, α ≥ 0, when the frequency of −1 is not in the interval ]14{1 \over 4}, 34{3 \over 4}[ We further obtain a generalization of Jensen-Jensen-Hoholdt’s result by establishing that the sequence of Littlewood polynomials is not L^α-flat for any α> 2 if the frequency of −1 is not 12{1 \over 2}. Finally, we prove that the sequence of palindromic Littlewood polynomials with even degrees are not L^α-flat for any α ≥ 0, and we provide a lemma on the existence of c-flat polynomials.

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K^×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓ^e is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such that the order of (G mod 𝔭) is congruent to a modulo ℓ_e only depends on a through its ℓ-adic valuation.

Let f (x) bea monicpolynomialwith integer coefficients and integers r₁,..., r_n with 0 ≤ r₁ ≤··· ≤ r_n <p the n roots of f (x) ≡ 0mod p for a prime p. We proposed conjectures on the distribution of the point (r₁/p,...,r_n/p) in the previous papers. One aim of this paper is to revise them for a reducible polynomial f (x), and the other is to show that they imply the one-dimensional equidistribution of r₁/p,...,r_n/p for an irreducible polynomial f (x) by a geometric way.

The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability distribution by a discrete distribution. Mixtures of probability distributions, also known as mixed distributions, are an exciting new area for optimal quantization. In this paper, we investigate the optimal quantization for three different mixed distributions generated by uniform distributions associated with probability vectors.