Number Theory Seminar: Matthew Smith

  • Date: 03/31/2011
  • Time: 16:10
Lecturer(s):
Matthew Smith
Location: 

Simon Fraser University

Topic: 

On additive combinatorics in higher degree systems

Description: 

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.

Other Information: 

Location: ASB 10900 (IRMACS - SFU Campus)

 

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