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# On Convolutions of Convex Sets and Related Problems

Published:2013-11-26
Printed: Dec 2014
• Tomasz Schoen,
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
 Format: LaTeX MathJax PDF

## Abstract

We prove some results concerning covolutions, the additive energy and sumsets of convex sets and its generalizations. In particular, we show that if a set $A=\{a_1,\dots,a_n\}_\lt \subseteq \mathbb R$ has the property that for every fixed $1\leqslant d\lt n,$ all differences $a_i-a_{i-d}$, $d\lt i\lt n,$ are distinct, then $|A+A|\gg |A|^{3/2+c}$ for a constant $c\gt 0.$
 Keywords: convex sets, additive energy, sumsets
 MSC Classifications: 11B99 - None of the above, but in this section

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