Improved Bound for Tomaszewski's Problem

Abstract

In 1986, Tomaszewski made the following conjecture. Given $n$ real numbers $a_1,...,a_n$ with ${\sum }_{i=1}^n{a}_i^2=1$, then of the $2^n$ signed sums $\pm a_1\pm ...\pm a_n$, at least half have absolute value at most $1$. Hendriks and Van Zuijlen (2020) and Boppana (2020) independently proved that a proportion of at least $0.4276$ of these sums has absolute value at most $1$. Using different techniques, we improve this bound to $0.46$.

12 pages