Extremal Problems About Asymptotic Bases: A Survey

I give an account of results and open problems on asymptotic bases in general additive number theory, inspired and arisen from the paper “On bases with an exact order” by Paul Erd˝os and Ronald L. Graham, published in Acta Arith. 37 (1980), 201–207. I survey papers by Melvyn B. Nathanson, Xing-De Jia, John C. M. Nash, Alain Plagne, Julien Cassaigne, Bruno Deschamps and myself

Estimation du nombre d'exceptions à ce qu'un ensemble de base privé d'un point reste un ensemble de base

International audienc

Comparison between lower and upper a-densities and lower and upper a-analytic densities

ABSTRACT. Let \alpha be a real number, with \alpha>-1. We prove a general in- equality between the upper (resp. lower) \alpha-analytic density and the upper (resp. lower) \alpha-density of a subset A of N^* (Proposition 2.1). Moreover, we prove by an example that the upper and the lower \alpha-densities and the lower and upper \alpha-analytic densities of A do not coincide in general (i.e., the inequalities proved in (2.1) may be strict). On the other hand, we identify a class of subsets of N^* for which these values do coincide in the case \alpha > -

Open Problems on Densities II

This is a collection of open questions and problems concerning various density concepts on subsets of N := {1,2,3,...}. It is a continuation of paper [10