# Probabilistic Littlewood-Offord anti-concentration results via model theory

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Abstract: (Joint with Jacob Fox and Matthew Kwan) The classical Erd*ő*s-Littlewood-Offord theorem says that for any n nonzero vectors in R^d, a random signed sum concentrates on any point with probability at most O(n^{1/2}). Combining tools from probability theory, additive combinatorics, and model theory, we obtain an anti-concentration probability of n^{-1/2+o(1)} for any o-minimal set S in R^d (such as a hypersurface defined by a polynomial in x1,...,xn,exp(x1),...,exp(xn), or a restricted analytic function) not containing a line segment. We do this by showing such o-minimal sets have no higher-order additive structure, complementing work by Pila on lower-order additive structure developed to count rational and algebraic points of bounded height.