On additive bases of sets with small product set

Abstract

We prove that finite sets of real numbers satisfying $|AA|\le |A{|}^{1+\epsilon }$ with sufficiently small $\epsilon >0$ cannot have small additive bases nor can they be written as a set of sums $B+C$ with $|B|,|C|\ge 2$. The result can be seen as a real analog of the conjecture of Sárközy that multiplicative subgroups of finite fields of prime order are additively irreducible.

arXiv comment10 pages, small corrections addressing reviewer's comments. To appear in Int. Math. Res. Notices