Pure Maths Colloquium: Alexia Yavícoli

The study of the relationship between the size of a set and the existence of arithmetic progressions contained in it has been a major problem for a long time. For example, Szemerédi’s theorem provides an answer in the discrete context: sets of natural numbers of positive density contain arithmetic progressions of all finite lengths. In the continuous context, one can study the same kind of problem: How large can a set of real numbers that avoids arithmetic progressions be? It is well known that sets of positive Lebesgue measure contain a homothetic copy of every finite set (in particular, arithmetic progressions of every finite length), so it is of interest to study the question using other notions of size. I will present some results in the continuous context providing answers to the previous question for various concepts of size, and also a result of the same type (joint with J. Fraser and P. Shmerkin) for approximate arithmetic progressions.