An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs

Abstract

An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs, Discrete Analysis 2020:12, 34 pp. The Littlewood-Offord problem is the following general question. Let $v_1,\dots,v_n$ be vectors in $\mathbb R^d$ of norm at least 1 and let $B\subset\mathbb R^d$ be a closed ball of diameter $\Delta$. Of the $2^n$ sums $\sum_{i\in E}v_i$, where $E\subset\{1,2,\dots,n\}$, how many can lie in $B$? Erdős observed that when $d=1$ and $\Delta Matthew Kwan talking about the results in this paper.