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arXiv · 2003.04075

An analytic approach to cardinalities of sumsets

Dávid Matolcsi·Imre Ruzsa·George Shakan·Dmitrii Zhelezov
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Abstract

Let $d$ be a positive integer and $U \subset \mathbb{Z}^d$ finite. We study $$\beta(U) : = \inf_{\substack{A , B \neq \emptyset \\ \text{finite}}} \frac{|A+B+U|}{|A|^{1/2}{|B|^{1/2}}},$$ and other related quantities. We employ tensorization, which is not available for the doubling constant, $|U+U|/|U|$. For instance, we show $$\beta(U) = |U|,$$ whenever $U$ is a subset of $\{0,1\}^d$. Our methods parallel those used for the Pr\'ekopa-Leindler inequality, an integral variant of the Brunn-Minkowski inequality.

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Dávid Matolcsi, Imre Ruzsa, George Shakan, Dmitrii Zhelezov. 2020-03-09. An analytic approach to cardinalities of sumsets. https://arxiv.org/abs/2003.04075

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