NSF FRG Analysis and Additive Combinatorics Working Group


This is an informal seminar series specifically aimed at graduate students. It is our intent to give expository lectures on a number of related results from analytic number theory, combinatorics and harmonic analysis (depending on the participants'/speakers' interests).

Unless stated otherwise our meeting will take place on Tuesdays from 3:30-5:00 in Boyd 410.

  • September 18, 2007
    • Speaker: Neil Lyall
    Title: Sarkozy's Theorem I, notes
    Abstract: We will discuss Sarkozy's theorem:
    If A is a subset of positive density in the integers, then there exists a and a' in A such that a-a' is a perfect square.
    The argument we plan to present, although elementary, in fact gives better quantitative bounds than those which were originally obtained by Sarkozy (but alas fall way short of the best bounds that are currently known). On the positive side, this argument is relatively simple and can be easily extended to prove a more general (and new) result on the existence of certain polynomial configurations in difference sets (or sumsets).

  • September 25, 2007
    • Speaker: Neil Lyall
    Title: Sarkozy's Theorem II, notes
    Abstract: We will continue our discussion of Sarkozy's theorem. We will recap the key points from last time, namely the estimates for Weyl sums, before focusing on the real heart of the proof:
    If A is "regular" or "pseudorandom" in the sense that it does not concentrated on any "long" square-difference arithmetic progression, then A necessarily contains "lots" of square differences.
  • October 2, 2007
    • Speaker: Neil Lyall
    Title: Brun's Combinatorial Sieve, notes, more notes
    Abstract: We will discuss Brun's combinatorial sieve and his proof that the sum of the reciprocals of the twin primes converges.
  • October 9, 2007
    • Speaker: Akos Magyar
    Title: Szemeredi's (unpublished) improvement of the bound in Roth's theorem I: The combinatorics
  • October 16, 2007
    Speaker: Akos Magyar
    Title: Szemeredi's (unpublished) improvement of the bound in Roth's theorem II: The Fourier analysis

  • November 6, 2007
    Speaker: Neil Lyall
    Title: An "energy increment stratagy" argument and polynomial configurations in sumsets
  • November 13, 2007
    Speaker: Akos Magyar
    Title: The U^3 uniformity norm inverse theorem I
  • November 27, 2007
    Speaker: Akos Magyar
    Title: The U^3 uniformity norm inverse theorem II
  • December 11, 2007
    Speaker:
    		 Matthew L. Smith
    Title:
    		 Difference sets without k-th powers