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Journal of the Institute of Mathematics of Jussieu (1)

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PRODUCT THEOREMS IN SL2 AND SL3
Mei-Chu Chang
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 7 / Issue 1 / January 2008

Published online by Cambridge University Press:

10 November 2006, pp. 1-25

Print publication:

January 2008
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We study product theorems for matrix spaces. In particular, we prove the following theorems.
Theorem 1. For all $\varepsilon>0$, there is $\delta>0$ such that if $A\subset\mathrm{SL}_3(\mathbb{Z})$ is a finite set, then either $A$ intersects a coset of a nilpotent subgroup in a set of size at least $|A|^{1-\varepsilon}$, or $|A^3|>|A|^{1+\delta}$.
Theorem 2. Let $A$ be a finite subset of $\mathrm{SL}_2(\mathbb{C})$. Then either $A$ is contained in a virtually abelian subgroup, or $|A^3|>c|A|^{1+\delta}$ for some absolute constant $\delta>0$.
Here $A^3=\{a_1a_2a_3:a_i\in A,\ i=1,2,3\}$ is the $3$-fold product set of $A$.