Tomaszewski's Problem on Randomly Signed sums, Revisited

Abstract

Let $v_1,v_2,\ldots, v_n$ be real numbers whose squares add up to 1.  Consider the $2^n$ signed sums of the form $S = \sum \pm v_i$.  Boppana and Holzman (2017) proved that at least 13/32 of these sums satisfy $|S| \le 1$.  Here we improve their bound to $0.427685$.