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