Proof of Tomaszewski's Conjecture on Randomly Signed Sums

Nathan Keller; Ohad Klein

Proof of Tomaszewski's Conjecture on Randomly Signed Sums

Combinatorics 2021-08-04 v4 Probability

Authors: Nathan Keller , Ohad Klein

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.

Keywords

probability theory

Cite

@article{arxiv.2006.16834,
  title  = {Proof of Tomaszewski's Conjecture on Randomly Signed Sums},
  author = {Nathan Keller and Ohad Klein},
  journal= {arXiv preprint arXiv:2006.16834},
  year   = {2021}
}