Multiplicative structures and random walks in o-minimal groups

We prove structure theorems for o-minimal definable subsets \(S\subset G\) $S\subset G$ 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 \(\mathbb {P}(X\in S)\le n^{-C}\) $\mathbb{P}(X\in S)\le n^{-C}$ and a structure theorem for the steps of X when \(\mathbb {P}(X\in S)\ge n^{-C'}\) $\mathbb{P}(X\in S)\ge n^{-C^{\prime }}$ .