Geometric and o-minimal Littlewood-Offord problems

Abstract: (Joint with Jacob Fox and Matthew Kwan, no o-minimal background required!) The classical Erdős-Littlewood-Offord theorem says that for any n nonzero vectors in $\mathbb{R}^d$, a random signed sum concentrates on any point with probability at most $O(n^{-1/2})$. Combining tools from probability theory, additive combinatorics, and o-minimality, we obtain an anti-concentration probability of $n^{-1/2+o(1)}$ for any o-minimal set $S$ in $\mathbb{R}^d$ (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.

computational geometrydiscrete mathematicscommutative algebracombinatorics

Audience: researchers in the topic

Copenhagen-Jerusalem Combinatorics Seminar

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