Combinatorics Seminar - Hunter Spink

16:15-18:00

Speaker: Hunter Spink

Title: Geometric and o-minimal Littlewood-Offord problems${\mathbb{}}^{}$

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 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 $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.