Proof of Tomaszewski's Conjecture on Randomly Signed Sums

TL;DR

This paper proves Tomaszewski's conjecture that for any sum of weighted random signs with normalized weights, the probability of the sum's absolute value being at most one is at least one-half, using advanced probabilistic inequalities.

Contribution

The paper provides a rigorous proof of Tomaszewski's conjecture employing novel local concentration inequalities and an improved Berry-Esseen inequality for Rademacher sums.

Findings

Proof confirms the conjecture for all cases.

Establishes new bounds using concentration inequalities.

Enhances understanding of Rademacher sum distributions.

Abstract

We prove the following conjecture, due to Tomaszewski (1986): Let $X= \sum_{i=1}^{n} a_{i} x_{i}$, where $\sum_i a_i^2=1$ and each $x_i$ is a uniformly random sign. Then $\Pr[|X|\leq 1] \geq 1/2$. Our main novel tools are local concentration inequalities and an improved Berry-Esseen inequality for Rademacher sums.