Arithmetic progressions in abundance by combinatorial tools

Abstract:

Using the algebraic structure of the Stone-Čech compactification of the integers, Furstenberg and Glasner proved that for arbitrary $k\in \mathbb {N}$, every piecewise syndetic set contains a piecewise syndetic set of $k$-term arithmetic progressions.

We present a purely combinatorial argument which allows us to derive this result directly from van der Waerden’s Theorem.