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On Convolutions of Convex Sets and Related Problems

Published online by Cambridge University Press:  20 November 2018

Tomasz Schoen*
Affiliation:
Faculty ofMathematics and Computer Science, AdamMickiewiczUniversity, Umultowska 87, 61-614 Poznań, Poland e-mail: schoen@amu.edu.pl
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Abstract

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We prove some results concerning convolutions, additive energies, and sumsets of convex sets and their generalizations. In particular, we show that if a set $A\,=\,{{\{{{a}_{1}},\,.\,.\,.\,,\,{{a}_{n}}\}}_{<}}\,\subseteq \,\mathbb{R}$ has the property that for every fixed $1\,\le \,d\,<\,n$, all differences ${{a}_{i}}\,-\,{{a}_{i-d}},\,d\,<\,i\,<n$, are distinct, then $\left| A\,+\,A \right|\,\gg \,{{\left| A \right|}^{3/2+c}}$ for a constant $c\,>\,0$.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2014

References

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