Category Archives: Topological Dynamics

Piecewise syndetic sets, topological dynamics and ultrafilters

In this post I explore the notion of piecewise syndeticity and its relation to topological dynamical systems and the Stone-Čech compactification. I restrict attention to the additive semigroup but most results presented are true in much bigger generality (and I … Continue reading

Posted in Classic results, Combinatorics, Tool, Topological Dynamics | Tagged , , | 1 Comment

Arithmetic progressions and the affine semigroup

— 1. Introduction — Van der Waerden’s theorem on arithmetic progressions states that, given any finite partition of , one of the cells contains arbitrarily long arithmetic progressions. For the sake of completion we give a precise formulation. Theorem 1 … Continue reading

Posted in Combinatorics, Topological Dynamics | Tagged , , | 1 Comment

Topological recurrence versus measure recurrence; a theorem of Kříž

— 1. Introduction — Previously on this blog I presented a proof of van der Waerden’s theorem which asserts that given a finite partition of , one of the cells of the partition contains arbitrarily long arithmetic progressions. It turns … Continue reading

Posted in Ergodic Theory, Ramsey Theory, Topological Dynamics | 3 Comments

Ratner’s Theorems

Ratner’s theorems are a series of results concerning unipotent flows of homogeneous spaces. They have been applied to many different situations, notably in some number theoretical questions, such as Oppenheim conjecture on quadratic forms. In this post I present the … Continue reading

Posted in Ergodic Theory, Tool, Topological Dynamics | Leave a comment

A topological proof of Van der Waerden’s Theorem

1. Introduction The van der Waerden’s theorem (vdW for short) was one of the first theorems to be proved in the branch of mathematics called Ramsey theory. I now state it in one of its many equivalent formulations: Theorem 1 … Continue reading

Posted in Ramsey Theory, Topological Dynamics | 13 Comments