On Additive Bases of Sets with Small Product Set

Journal article, 2018

We prove that finite sets of real numbers satisfying vertical bar AA vertical bar <= vertical bar A vertical bar (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 vertical bar B vertical bar, vertical bar C vertical bar >= 2. The result can be seen as a real analog of the conjecture of Sarkozy that multiplicative subgroups of finite fields of prime order are additively irreducible.