Additive properties of sequences of pseudo s-th powers

TL;DR

This paper investigates the additive properties of sequences of pseudo s-th powers, showing they almost surely form bases of order s + x and analyzing the threshold phenomena for additive complements.

Contribution

It establishes that such sequences are almost surely bases of order s + x and characterizes the threshold behavior for additive complements.

Findings

Sequences are almost surely bases of order s + x for any x > 0.

Threshold phenomena occur in the size of additive complements.

Precise theorems describe the minimal size of additive complements.

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.