Applications of the sum-product theorem in finite fields

Summary form only given. About two years ago Bourgain, Katz and Tao (2004) proved the following theorem, essentially stating that in every finite field, a set which does not grow much when we add all pairs of elements, and when we multiply all pairs of elements, must be very close to a subfield. Theorem 1: (Bourgain et al., 2004) For every epsi > 0 there exists a delta > 0 such that the following holds. Let F be any field with no subfield of size ges |F|epsi. For every set A sube F, with |F|epsi < |A| < |F|1 - epsi, either the sumset |A + A| > |A|1 + delta or the product set |A times A| > |A|1 + delta. This theorem revealed its fundamental nature quickly. Shortly afterwards it has found many diverse applications, including in number theory, group theory, combinatorial geometry, and the explicit construction of extractors and Ramsey graphs, mostly described in the references below. In my talk I plan to explain some of the applications, as well as to sketch the main ideas of the proof of the sum-product theorem