Ramsey theory for p-quasicyclic groups with a view towards topological dynamics

TL;DR

This paper develops additive and multiplicative partition theorems for p-quasicyclic groups, explores density and recurrence properties, and extends topological dynamics concepts to these groups, revealing new combinatorial and dynamical insights.

Contribution

It introduces the first combinatorial and dynamical results for p-quasicyclic groups, extending classical theorems to this new setting.

Findings

Proved additive and multiplicative partition theorems for p-quasicyclic groups.

Established density results using F{46}lner sequences for these groups.

Extended topological dynamical systems and proved recurrence theorems analogous to classical results.

Abstract

We prove additive and multiplicative partition theorems, obtaining combinatorial results for p-quasicyclic groups, where p is a prime number. We also get density results for p-quasicyclic groups via left F{\o}lner sequences of non-empty finite subsets of it, giving a sufficient condition in order a subset of a p-quasicyclic group to contain arbitrary long arithmetic progressions. Finally, we introduce the notion of a dynamical system over p-quasicyclic groups extending the classical notion of a topological dynamical system and we prove (multiple) recurrent results for the p-quasicyclic groups. In particular, we prove recurrent results analogous to Furstenberg-Weiss type theorems for classical systems.