The sum-product theorem and applications
Abstract
How "orthogonal" are the basic field operations "+" and "x"?
About two years ago Bourgain, Katz and Tao proved the following
theorem (stated very informally). 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. In particular, prime fields have no such subsets!
This theorem revealed its fundamental nature quickly. Shortly
afterwards it has found many diverse applications, including in
Number Theory, Analysis, Group Theory, Combinatorial Geometry, and the
explicit construction of Randomness Extractors and Ramsey Graphs.
In this 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.