Submission date: 9 June 2021

Abstract:

The classical Erdős-Littlewood-Offord theorem says that for nonzero vectors a_1,...,a_n in ℝ^d, any x in ℝ^d, and uniformly random (ξ_1,...,ξ_n) in {-1,1}^n, we have Pr(a_1ξ_1+...+a_nξ_n=x)=O(n^{-1/2}). In this paper we show that Pr(a_1ξ_1+...+a_nξ_n in S) ≤ 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.

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Full text arXiv 2106.04894: pdf, ps.