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arXiv · 1502.03700

On sets with small additive doubling in product sets

Dmitry Zhelezov
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Abstract

Following the sum-product paradigm, we prove that for a set $B$ with polynomial growth, the product set $B.B$ cannot contain large subsets with size of order $|B|^2$ with small doubling. It follows that the additive energy of $B.B$ is asymptotically $o(|B|^6)$. In particular, we extend to sets of small doubling and polynomial growth the classical Multiplication Table theorem of Erd\H{o}s saying that $|[1..n]. [1..n]| = o(n^2)$.

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Dmitry Zhelezov. 2015-02-12. On sets with small additive doubling in product sets. https://arxiv.org/abs/1502.03700

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