Multiplicative structures and random walks in o-minimal groups

TL;DR

This paper investigates the structure of definable subsets in o-minimal groups with large multiplicative structures, proving new theorems about torsion and providing bounds for random walks in these groups.

Contribution

It establishes structure theorems for o-minimal definable sets with multiplicative structures and analyzes random walk behaviors in definable groups.

Findings

Definable groups lack bounded torsion near the identity.

Upper bounds on probabilities for random walks hitting certain sets.

Structure theorems for steps of random walks when hitting probabilities are large.

Abstract

We prove structure theorems for o-minimal definable subsets $S\subset G$ of definable groups containing large multiplicative structures, and show definable groups do not have bounded torsion arbitrarily close to the identity. As an application, for certain models of $n$-step random walks $X$ in $G$ we show upper bounds $\mathbb{P}(X\in S)\le n^{-C}$ and a structure theorem for the steps of $X$ when $\mathbb{P}(X\in S)\ge n^{-C'}$.