Geometric and o-minimal Littlewood-Offord problems

The classical ErdH{o}s-Littlewood-Offord theorem says that for nonzero vectors $a_1,dots,a_nin mathbb{R}^d$, any $xin mathbb{R}^d$, and uniformly random $(xi_1,dots,xi_n)in{-1,1}^n$, we have $Pr(a_1xi_1+dots+a_nxi_n=x)=O(n^{-1/2})$. In this paper we show that $Pr(a_1xi_1+dots+a_nxi_nin S)le n^{-1/2+o(1)}$ whenever $S$ is definable with respect to an o-minimal structure (for example, this holds when $S$ is any algebraic hypersurface), under the necessary condition that it does not contain a line segment. We also obtain an inverse theorem in this setting.