Number Theory Seminar: Matthew Smith
Topic
On additive combinatorics in higher degree systems
Speakers
Details
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.
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.
Additional Information
Location: ASB 10900 (IRMACS - SFU Campus)
For more information please visit UBC Mathematics department
Matthew Smith