The sum-product theorem and its applications
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 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, 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.