Probabilistic Littlewood-Offord anti-concentration results via model theory

Abstract: Abstract: (Joint with Jacob Fox and Matthew Kwan) 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 $x_1,...,x_n,e^{x_1},...,e^{x_n}$, 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.

combinatoricslogicprobability

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic