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Large values of the additive energy in ${\mathbb{R}^d$ and ${\mathbb{Z}^d$

Published online by Cambridge University Press:  09 January 2014

XUANCHENG SHAO
XUANCHENG SHAO*
Affiliation:
Department of Mathematics, Stanford University450 Serra Mall, Bldg. 380, Stanford, CA, 94305-2125, U.S.A. e-mail: xshao@math.stanford.edu
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Abstract

Combining Freiman's theorem with Balog–Szemerédi–Gowers theorem one can show that if an additive set has large additive energy, then a large piece of the set is contained in a generalized arithmetic progression of small rank and size. In this paper, we prove the above statement with the optimal bound for the rank of the progression. The proof strategy involves studying upper bounds for additive energy of subsets of ${\mathbb{R}^d$ and ${\mathbb{Z}^d$.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 2 , March 2014 , pp. 327 - 341
Copyright
Copyright © Cambridge Philosophical Society 2014 

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