On additive bases of sets with small product set

TL;DR

This paper demonstrates that finite sets of real numbers with very small multiplicative growth cannot be efficiently expressed as additive bases or simple sumsets, highlighting a fundamental incompatibility between small product sets and additive structure.

Contribution

It establishes a new result linking small multiplicative doubling to the impossibility of small additive bases in real sets, extending ideas analogous to finite field conjectures.

Findings

Sets with small |AA| cannot have small additive bases

Such sets cannot be written as B+C with |B|, |C| ≥ 2

Results extend multiplicative-additive incompatibility concepts

Abstract

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