Abstract view

# 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
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
 Format: LaTeX MathJax PDF

## 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