Mathematical Conversations

It turns out that certain additive patterns in the integers are very hard to get rid of. An instance of this is captured in a conjecture of Erdős, which states that as long as a set of natural numbers is 'somewhat dense' -- namely the sum of the reciprocals of its elements diverges -- the set must contain arbitrarily long arithmetic progressions. In this talk, I'll discuss recent work with Bloom where we prove the simplest case of the conjecture: finding arithmetic progressions of length 3. This case has a rich history, and our work builds particularly upon that of Roth, Bourgain and Bateman--Katz. I'll aim to showcase some of this history, and in particular the problem's lovely connections with harmonic analysis and convex geometry.