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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
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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