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On the size of $k$-fold sum and product sets of integers

Jean Bourgain·Mei-Chu Chang
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Abstract

We prove the following theorem: for all positive integers $b$ there exists a positive integer $k$, such that for every finite set $A$ of integers with cardinality $|A| > 1$, we have either $$ |A + ... + A| \geq |A|^b$$ or $$ |A \cdot ... \cdot A| \geq |A|^b$$ where $A + ... + A$ and $A \cdot ... \cdot A$ are the collections of $k$-fold sums and products of elements of $A$ respectively. This is progress towards a conjecture of Erdös and Szemerédi on sum and product sets.

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Jean Bourgain, Mei-Chu Chang. 2003-09-03. On the size of $k$-fold sum and product sets of integers. https://arxiv.org/abs/math/0309055

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