Sum Product Theorem

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.

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There are two labyrinths of the human mind: one concerns the composition of the continuum, and the other the nature of freedom, and both spring from the same source—the infinite. —Baron von Leibniz...

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...