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, 1n, we have Pr (a1ξ1+⋯+anξn=x) =O (n−1/2). In this paper, we show that Pr (a1ξ1+⋯+anξn∈S) ≤n−1/2+o (1) whenever S is definable with respect to an o-minimal structure (e. g. , 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.