## Number Theory Seminar: Matthew Smith

- Date: 03/31/2011
- Time: 16:10

Simon Fraser University

On additive combinatorics in higher degree systems

Abstract

We consider a system of k diagonal polynomials of degrees 1, 2,..., k.

Using methods developed by W.T. Gowers and refined by Green and Tao to

obtain bounds in the 4-term case of Szemeredi's Theorem on long

arithmetic progressions, we show that if a subset A of the natural

numbers up to N of size d_N*N exhibits sufficiently small local

polynomial bias, then it furnishes roughly the expected number of

solutions to the given system. If A furnishes no non-trivial solutions

to the system, then we show via an energy incrementing argument that

there is a concentration in a Bohr set of pure degree k, and

consequently in a long arithmetic progression. We show that this leads

to a bound on the density d_N of the set A of the form d_N <<

exp(-c*sqrt(log log N)), where c>0 is a constant dependent at most on

k.

Location: ASB 10900 (IRMACS - SFU Campus)

For more information please visit UBC Mathematics department