# Category Archives: guest blogging

## Growth, expanders, and the sum-product problem

It is a great pleasure to introduce Harald Helfgott as the first guest author on this blog.  Many of the readers here will be familiar with the sum-product problem of Erdös-Szemerédi: see here for general background, or here for an exposition of Solymosi’s recent breakthrough. The subject of this post is the connection between the sum-product theorem in finite fields, due to Bourgain-Katz-Tao and Konyagin, and recent papers on sieving and expanders, including Bourgain and Gamburd’s papers on $SL_2$ and on $SL_3$ and Bourgain, Gamburd and Sarnak’s paper on the affine sieve. Since this connection was made through Harald’s paper on growth in $SL_2$ (he has since proved a similar result in $SL_3$ ), I asked Harald if he would be willing to write about it here. His post is below.

HARALD HELFGOTT:

Let us first look at the statements of the main results. In the following, $|S|$ means “the number of elements of a set $S$“. Given a subset $A$ of a ring, we write $A+A$ for $\{x+y:\ x,y \in A\}$ and $A\cdot A$ for $\{x\cdot y:\ x,y \in A\}$.

Sum-product theorem for ${\Bbb Z}/p{\Bbb Z}$ (Bourgain, Katz, Tao, Konyagin). Let $p$ be a prime. Let $A$ be a subset of ${\Bbb Z}/p{\Bbb Z}$. Suppose that $|A|\leq p^{1-\epsilon},\ \epsilon>0$. Then either $|A+A| \geq |A|^{1+\delta}$ or $|A\cdot A|\geq |A|^{1+\delta}$, where $\delta>0$ depends only on $\epsilon$.

In other words: a subset of ${\Bbb Z}/p{\Bbb Z}$ grows either by addition or by multiplication (provided it has any room to grow at all). One of the nicest proofs of the sum-product theorem can be found in this paper by Glibichuk and Konyagin.

Now here is what I proved on $SL_2$. Here $A\cdot A\cdot A$ means simply $\{x\cdot y\cdot z:\ x,y,z\in A\}$.

Theorem (H). Let $p$ be a prime. Let $G = SL_2({\Bbb Z}/p{\Bbb Z})$. Let $A$ be a subset of $G$ that generates $G$. Suppose that $|A|\leq |G|^{1-\epsilon},\ \epsilon>0$. Then $|A\cdot A\cdot A| \geq |A|^{1+ \delta}$, where $\delta >0$ depends only on $\epsilon$.