The quantum group fixing a sequence of finite subsets

TL;DR

This paper generalizes Szemerédi's theorem within the framework of discrete quantum groups, showing that elements fixing a sequence of finite subsets form a quantum subgroup and establishing a mean ergodic theorem for these groups.

Contribution

It introduces a novel quantum group analogue of Szemerédi's theorem and proves the quantum subgroup structure of elements fixing finite subsets.

Findings

Elements fixing finite subsets form a quantum subgroup

Established a mean ergodic theorem for discrete quantum groups

Extended classical combinatorial theorems to quantum group setting

Abstract

Motivated by generalizing Szemer\'edi's theorem, we the elements in a discrete quantum group fixing a sequence of finite subsets and prove that the set of these elements is a quantum subgroup. Using this we obtain a version of mean ergodic theorem for discrete quantum groups.