Counting progressions in the IP Szemerédi Theorem

Szemerédi's theorem (1975) asserts that any subset of the integers with positive density contains arbitrarily long arithmetic progressions. A famous generalization of this result is the IP Szemerédi theorem, established by Furstenberg and Katznelson (1985). This theorem restricts the common differences of these progressions to an IP. The term IP stands for Infinite (dimensional) Parallelepiped and is defined as the collection of all finite sums of distinct terms from a given sequence of integers.

 

While these classical results tell us that such patterns exist, they do not tell us how many appear. In this talk, I will outline the history of these theorems and present a new perspective that bridges this gap.