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arXiv · 1407.5291

Additive properties of sequences of pseudo s-th powers

Javier Cilleruelo·Jean-Marc Deshouillers·Victor Lambert·Alain Plagne
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Abstract

In this paper, we study (random) sequences of pseudo s-th powers, as introduced by Erdös and Rényi 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.

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Javier Cilleruelo, Jean-Marc Deshouillers, Victor Lambert, Alain Plagne. 2014-07-20. Additive properties of sequences of pseudo s-th powers. https://arxiv.org/abs/1407.5291

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