Proof of Tomaszewski's Conjecture on Randomly Signed Sums

Nathan Keller; Ohad Klein

paper

Proof of Tomaszewski's Conjecture on Randomly Signed Sums

arXiv:2006.16834

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.

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