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

## TL;DR

This paper improves the lower bound on the proportion of signed sums of real numbers with unit squared sum that are within 1 in absolute value, surpassing the previous 3/8 barrier using a novel method.

## Contribution

The authors introduce a new approach that improves the known bound from 3/8 to 13/32 for the fraction of signed sums within a certain bound.

## Key findings

- Improved the lower bound from 3/8 to 13/32.
- Established a new method surpassing previous limitations.
- Demonstrated the effectiveness of the novel approach.

## Abstract

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 $|S| \le 1$. 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.