Improved Bound for Tomaszewski's Problem

Abstract

In 1986, Tomaszewski made the following conjecture. Given $n$ real numbers $a_{1},\ldots,a_{n}$ with $\sum_{i=1}^{n}a_{i}^{2}=1$, then of the $2^{n}$ signed sums $\pm a_{1} \pm \cdots \pm a_{n}$, at least half have absolute value at most 1. Hendriks and van Zuijlen [An Improvement of the Boppana-Holzman Bound for Rademacher Random Variables}, arXiv:2003.02588, 2020] and Boppana, Hendriks, and van Zuijlen [Tomaszewski's Problem on Randomly Signed Sums, Revisited, arXiv:2003.06433, 2020] independently proved that a proportion of at least 0.4276 of these sums has absolute value at most 1. Using different techniques, we improve this bound to 0.46.

Locations

• SIAM Journal on Discrete Mathematics - View
• arXiv (Cornell University) - View - PDF