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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 Liu and Spencer

We describe the structure of ‘K-approximate subgroups’ of torsion-free nilpotent groups, paying particular attention to Lie groups.

Three other works, by Fisher et al., by Sanders and by Tao, have appeared that independently address related issues. We comment briefly on some of the connections between these papers.

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Structure in Sets with Logarithmic Doubling

Approximate groups. I The torsion-free nilpotent case

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2 results

Structure in Sets with Logarithmic Doubling

Approximate groups. I The torsion-free nilpotent case