Tomaszewski's Problem on Randomly Signed Sums: Breaking the 3/8 Barrier

Let $v_1$, $v_2$, …, $v_n$ be real numbers whose squares add up to 1. Consider the $2^n$ signed sums of the form $S=\sum \pm v_i$. Holzman and Kleitman (1992) proved that at least 3/8 of these sums satisfy $\mathrm{\mid }S\mathrm{\mid }\le 1$. This 3/8 bound seems to be the best their method can achieve. Using a different method, we improve the bound to 13/32, thus breaking the 3/8 barrier.