ADDITIVE PROPERTIES OF SEQUENCES OF PSEUDO s-TH POWERS

In this paper, we study (random) sequences of pseudo s-th powers, as introduced by Erdős 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 + for any > 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.