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A Note on Freĭman's Theorem in Vector Spaces

Published online by Cambridge University Press:  01 March 2008

T. SANDERS
T. SANDERS*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK (e-mail: [email protected])
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Abstract

A famous result of Freĭman describes the sets A, of integers, for which |A+A| ≤ K|A|. In this short note we address the analogous question for subsets of vector spaces over . Specifically we show that if A is a subset of a vector space over with |A+A| ≤ K|A| then A is contained in a coset of size at most 2O(K3/2 log K)|A|, which improves upon the previous best, due to Green and Ruzsa, of 2O(K2)|A|. A simple example shows that the size may need to be at least 2Ω(K)|A|.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 2 , March 2008 , pp. 297 - 305
Copyright
Copyright © Cambridge University Press 2007

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