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Structure in Sets with Logarithmic Doubling
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- Journal:
- Canadian Mathematical Bulletin / Volume 56 / Issue 2 / 01 June 2013
- Published online by Cambridge University Press:
- 20 November 2018, pp. 412-423
- Print publication:
- 01 June 2013
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- Article
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Suppose that
$G$ is an abelian group, 
$A\,\subset \,G$ is finite with 
$\left| A\,+\,A \right|\,\le \,K\left| A \right|$ and 
$\eta \,\in \,(0,\,1]$ is a parameter. Our main result is that there is a set 
$L$ such that
$$\left| A\,\cap \,\text{Span}\left( L \right) \right|\ge {{K}^{-{{O}_{n}}\left( 1 \right)}}\left| A \right|\,\,\,\,\text{and}\,\,\,\,\,\left| L \right|=O\left( {{K}^{n}}\log \left| A \right| \right).$$We include an application of this result to a generalisation of the Roth-Meshulam theorem due to Liuand Spencer
