Sets of integers containing (almost) arithmetic progressions

Abstract

An arithmetic progression is a finite sequence of real numbers with a constant gap sequence. For example, 1,3,5,7 is an arithmetic progression of length four (there are four terms) and gap size two (the gap between any two consecutive terms is two). Given a set of integers, it is natural to ask whether or not it contains arithmetic progressions of certain lengths.

The Erdos conjecture on arithmetic progressions says that if the reciprocals of a set of positive integers form a divergent series then the set should contain arbitrarily long arithmetic progressions, that is, for all k, there is an arithmetic progression of length k somewhere inside the set. The celebrated (and deep) Green-Tao Theorem is a special case of this conjecture. Jonathan will discuss a simple approach to this type of problem where one can show that such sets get arbitrarily close to arbitrarily long arithmetic progressions.