arXiv subjects

Georges Grekos

Publications and source records attributed to Georges Grekos.

4 recordsLinked to original sources

arXiv2024

The Beurling-Malliavin density, the Polya density and their connection

In this paper we present a new formulation of the Beurling-Malliavin density (Proposition 1). Then we consider the upper Polya density and show how its existence is connected with the concept of subadditivity; moreover, by means of some quantities introduced for proving Proposition 1, a theorem is presented that clarifies the connection between the upper Polya and the Beurling-Malliavin densities. In the last section we discuss the classical definition of the upper Polya density and we prove a result which seems to be new.

math.NT↗

arXiv2023

On the connection between the Beurling-Malliavin density and the asymptotic density

We study the notion of Beurling-Malliavin density from the point of view of Number Theory. We prove a general relation between the Beurling-Malliavin density and the upper asymptotic density; we identify a class of sequences for which the two densities coincide; this class contains the arithmetic progressions. Last, by means of an alternative definition of Beurling-Malliavin density, we study the connection with the asymptotic density for another kind of sequences that again generalizes the arithmentic progressions.

math.NT↗

arXiv2019

Additive Complements for a given Asymptotic Density

{The first version of this text was written and submitted to a journal on April, 12, 2018. This second version was submitted on April, 9, 2019.} We investigate the existence of subsets $A$ and $B$ of $\mathbb{N}:=\{0,1,2,\dots\}$ such that the sumset $A+B:=\{a+b~;a\in A,b\in B\}$ has given asymptotic density. We solve the particular case in which $B$ is a given finite subset of $\mathbb{N}$ and also the case when $B=A$ ; in the later case, we generalize our result to $kA:=\{x_1+\cdots+x_k: x_i\in A, i=1,\dots,k\}$ for an integer $k\geq2.$

math.NT↗

arXiv2012

Gaps and the exponent of convergence of an integer sequence

Professor Tibor Šalát, at one of his seminars at Comenius University, Bratislava, asked to study the influence of gaps of an integer sequence A={a_1<a_2<...<a_n<...} on its exponent of convergence. The exponent of convergence of A coincides with its upper exponential density. In this paper we consider an extension of Professor Šalát's question and we study the influence of the sequence of ratios a_m/a_{m+1} and of the sequence (a_{m+1}-a_m)/a_m on the upper and on the lower exponential densities of A.

math.NT↗