# Approximate Subgroups of Linear Groups

@article{Breuillard2010ApproximateSO,
title={Approximate Subgroups of Linear Groups},
author={Emmanuel Breuillard and Ben Green and Terence Tao},
journal={Geometric and Functional Analysis},
year={2010},
volume={21},
pages={774-819}
}
• Published 11 May 2010
• Mathematics
• Geometric and Functional Analysis
We establish various results on the structure of approximate subgroups in linear groups such as SLn(k) that were previously announced by the authors. For example, generalising a result of Helfgott (who handled the cases n = 2 and 3), we show that any approximate subgroup of $${{\rm SL}_{n}({\mathbb {F}}_{q})}$$ which generates the group must be either very small or else nearly all of $${{\rm SL}_{n}({\mathbb {F}}_{q})}$$. The argument generalises to other absolutely almost simple connected (and…
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## References

SHOWING 1-10 OF 78 REFERENCES

### Linear Approximate Groups

• Mathematics
• 2010
This is an informal announcement of results to be described and proved in detail in a paper to appear. We give various results on the structure of approximate subgroups in linear groups such as

### Affine linear sieve, expanders, and sum-product

• Mathematics
• 2010
AbstractLet $\mathcal{O}$ be an orbit in ℤn of a finitely generated subgroup Λ of GLn(ℤ) whose Zariski closure Zcl(Λ) is suitably large (e.g. isomorphic to SL2). We develop a Brun combinatorial

### A Strong Tits Alternative

We show that for every integer $d$, there is a constant $N(d)$ such that if $K$ is any field and $F$ is a finite subset of $GL_d(K)$, which generates a non amenable subgroup, then $F^{N(d)}$ contains

### Congruence Properties of Zariski‐Dense Subgroups I

• Mathematics
• 1984
This paper deals with the following general situation: we are given an algebraic group G defined over a number field K, and a subgroup F of the group G(K) of K-rational points of G. Then what should

### Product decompositions of quasirandom groups and a Jordan type theorem

• Mathematics
• 2007
We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G, then for every subset B of G

### Stable group theory and approximate subgroups

We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite

### The Kakeya set and maximal conjectures for algebraic varieties over finite fields

• Mathematics
• 2009
Using the polynomial method of Dvir \cite{dvir}, we establish optimal estimates for Kakeya sets and Kakeya maximal functions associated to algebraic varieties $W$ over finite fields $F$. For

### A Finitary Version of Gromov’s Polynomial Growth Theorem

• Mathematics
• 2009
We show that for some absolute (explicit) constant C, the following holds for every finitely generated group G, and all d > 0: If there is someR0 > exp(exp(CdC)) for which the number of elements in a

### PRODUCT THEOREMS IN SL2 AND SL3

• Mei-Chu Chang
• Mathematics
Journal of the Institute of Mathematics of Jussieu
• 2006
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 $A\subset\mathrm{SL}_3(\mathbb{Z})$

### On the minimal degrees of projective representations of the finite Chevalley groups

• Mathematics
• 1974
For G = G(q), a Chevalley group defined over the field iFQ of characteristic p, let Z(G,p) be th e smallest integer t > 1 such that G has a projective irreducible representation of degree t over a