Additive properties of sequences of pseudo s-th powers
Number Theory
2014-07-22 v1
Abstract
In this paper, we study (random) sequences of pseudo s-th powers, as introduced by Erd\"os and R\'enyi in 1960. In 1975, Goguel proved that such a sequence is almost surely not an asymptotic basis of order s. Our first result asserts that it is however almost surely a basis of order s + x for any x > 0. We then study the s-fold sumset sA = A + ... + A (s times) and in particular the minimal size of an additive complement, that is a set B such that sA + B contains all large enough integers. With respect to this problem, we prove quite precise theorems which are tantamount to asserting that a threshold phenomenon occurs.
Keywords
Cite
@article{arxiv.1407.5291,
title = {Additive properties of sequences of pseudo s-th powers},
author = {Javier Cilleruelo and Jean-Marc Deshouillers and Victor Lambert and Alain Plagne},
journal= {arXiv preprint arXiv:1407.5291},
year = {2014}
}