An example of a basic and powerful theorem in arithmetic combinatorics is the sum product theorem of Jean Bourgain, Nets Katz, and Terence Tao. It is an elementary but fundamental quantitative combinatorial fact about the way addition and multiplication work in finite sets of integers. Its generalizations have wide applications to algebra, number theory, theoretical computer science, and most recently to group theory.
During the first term of 2007–08, School of Mathematics
Professor Jean Bourgain and Member Van Vu of Rutgers, The State
University of New Jersey, ran a program on arithmetic
combinatorics. The Members in residence for the program ranged from