David Conlon: A relative Szemerédi theorem

Lunch is served at 12:00 noon (register at this doodle by Sunday April 6 at 8 pm). The presentation starts at 12:10 pm and ends at 1 pm. Those of us who wish reconvene after a short break for ca two hours of more technical discussions.

Abstract

The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemerédi theorem which says that any subset of a pseudorandom set of integers of positive relative density contains long arithmetic progressions.

In this talk, we will discuss a simple proof of a strengthening of the relative Szemerédi theorem, showing that a much weaker pseudorandomness condition is sufficient. Our strengthened version can be applied to give the first relative Szemerédi theorem for k-term arithmetic progressions in pseudorandom subsets of Z_N of density N^{-c_k}.

The key component in our proof is an extension of the regularity method to sparse pseudorandom hypergraphs, which we believe to be interesting in its own right. From this we derive a relative extension of the hypergraph removal lemma. This is a strengthening of an earlier theorem used by Tao in his proof that the Gaussian primes contain arbitrarily shaped constellations and, by standard arguments, allows us to deduce the relative Szemerédi theorem.

This is joint work with Jacob Fox and Yufei Zhao.