Multiplicative structures and random walks in o-minimal groups

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 \(\mathbb {P}(X\in S)\ge n^{-C'}\).