Additive properties of sequences of pseudo s-th powers

作者：Cilleruelo Javier; Deshouillers Jean Marc^*; Lambert Victor; Plagne Alain

来源：Mathematische Zeitschrift, 2016, 284(1-2): 175-193.

DOI：10.1007/s00209-016-1651-8

• Additive basis
• Pseudo s-powers
• Probabilistic method
• Additive number theory

In this paper, we study (random) sequences of pseudo s-th powers, as introduced by ErdAs and R,nyi (Acta Arith 6:83-110, 1960). Goguel (J Reine Angew Math 278/279:63-77, 1975) 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 for any . We then study the s-fold sumset (s times) and in particular the minimal size of an additive complement, that is a set B such that 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.

• 出版日期2016-10