Probabilistic Littlewood–Offord anti-concentration results via model theory

The classical Erdos–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 structures developed to count rational and algebraic points of bounded height.

This is joint work with Jacob Fox and Matthew Kwan.