Geometric and o-minimal Littlewood–Offord problems

The classical Erdős–Littlewood–Offord theorem says that for nonzero vectors a1,…,an∈Rd, any x∈Rd, and uniformly random (ξ1,…,ξn)∈{−1,1}n, we have Pr(a1ξ1+⋯+anξn=x)=O(n−1/2). In this paper, show Pr(a1ξ1+⋯+anξn∈S)≤n−1/2+o(1) whenever S is definable with respect to an o-minimal structure (e.g., holds when algebraic hypersurface), under the necessary condition it does not contain a line segment. We also obtain inverse in setting.