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arXiv · 1606.02320

On additive bases of sets with small product set

Ilya D. Shkredov·Dmitrii Zhelezov
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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.

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Ilya D. Shkredov, Dmitrii Zhelezov. 2016-06-07. On additive bases of sets with small product set. https://arxiv.org/abs/1606.02320

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