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Structure in Sets with Logarithmic Doubling

  Published:2011-09-15
 Printed: Jun 2013
  • T. Sanders,
    Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK
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Abstract

Suppose that $G$ is an abelian group, $A \subset G$ is finite with $|A+A| \leq K|A|$ and $\eta \in (0,1]$ is a parameter. Our main result is that there is a set $\mathcal{L}$ such that \begin{equation*} |A \cap \operatorname{Span}(\mathcal{L})| \geq K^{-O_\eta(1)}|A| \quad\text{and}\quad |\mathcal{L}| = O(K^\eta\log |A|). \end{equation*} We include an application of this result to a generalisation of the Roth--Meshulam theorem due to Liu and Spencer.
Keywords: Fourier analysis, Freiman's theorem, capset problem Fourier analysis, Freiman's theorem, capset problem
MSC Classifications: 11B25 show english descriptions Arithmetic progressions [See also 11N13] 11B25 - Arithmetic progressions [See also 11N13]
 

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