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 and on and Bourgain, Gamburd and Sarnak’s paper on the affine sieve. Since this connection was made through Harald’s paper on growth in (he has since proved a similar result in ), I asked Harald if he would be willing to write about it here. His post is below.
Let us first look at the statements of the main results. In the following, means “the number of elements of a set “. Given a subset of a ring, we write for and for .
Sum-product theorem for (Bourgain, Katz, Tao, Konyagin). Let be a prime. Let be a subset of . Suppose that . Then either or , where depends only on .
In other words: a subset of 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 . Here means simply .
Theorem (H). Let be a prime. Let . Let be a subset of that generates . Suppose that . Then , where depends only on .