EUDML

We introduce the concept of uniform weighted density (upper and lower) of a subset $A$ of $ℕ^{*}$ , with respect to a given sequence of weights $(a_{n})$ . This concept generalizes the classical notion of uniform density (for which the weights are all equal to 1). We also prove a theorem of comparison between two weighted densities (having different sequences of weights) and a theorem of comparison between a weighted uniform density and a weighted density in the classical sense. As a consequence, new bounds for the...

Regular sets and conditional density: an extension of Benford's law

Rita Giuliano Antonini; Georges Grekos — 2005

Colloquium Mathematicae

We give an extension of Benford's law (first digit problem) by using the concept of conditional density, introduced by Fuchs and Letta. The main tool is the notion of regular subset of integers.

A note on uniform or Banach density

Georges Grekos; Vladimír Toma; Jana Tomanová — 2010

Annales mathématiques Blaise Pascal

In this note we present and comment three equivalent definitions of the so called or density of a set of positive integers.

Some generalizations of Olivier's theorem

Alain Faisant; Georges Grekos; Ladislav Mišík — 2016

Mathematica Bohemica

Let $\sum_{n = 1}^{\infty} a_{n}$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $(a_{n})$ is non-increasing, then $lim_{n \to \infty} n a_{n} = 0$ . In the present paper: (a) We formulate and prove a necessary and sufficient condition for having $lim_{n \to \infty} n a_{n} = 0$ ; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the $ℐ$ -convergence, that is a convergence according to an ideal $ℐ$ of subsets of $ℕ$ . Again, Olivier’s theorem is a consequence of...