Some problems in additive combinatorics

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

The proposer plans to work on a variety of problems in additive combinatorics, specifically: Continuing his work on the Polymath4 project to produce a deterministic algorithm to find large prime numbers quickly; exploring the ramifications of a ground-breaking probabilistic approach that he has developed to attack certain additive combinatorial problems (such as the 2D corners problem); and continuing the development of some fresh ideas on sum-product inequalities. A good example here is the proposer's work in the Polymath4 project: To date he, and other participants, have produced an algorithm to compute the parity of the prime counting function faster than any previous method; and furthermore, the proposer has produced a ``polynomial ring analogue'' of this algorithm which shows great promise towards completing the project (when generalized further, perhaps). Understanding the structure of prime numbers (2,3,5,7,11, and so) is one of the enduring legacies of the ancient Greeks, and in recent years enormous progress has been made. Polymath4, an online collaborative research program initiated by Timothy Gowers, Gil Kalai, and Terrence Tao, addresses a related, fundamental question: How quickly can one even produce large prime numbers? The proposer has been a major participant in this project, and has thus far made significant strides in addressing this problem. He plans to continue working on it with other Polymath4 researchers and with some of his students. In addition, he plans to continue his work on developing some ground-breaking methods in a relatively new field called ``additive combinatorics'', which has become a hot area of late due to works of Gowers, Green, Tao, and others.