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arXiv · 2005.05031

Improved Bound for Tomaszewski's Problem

Vojtěch Dvořák·Peter van Hintum·Marius Tiba
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Abstract

In 1986, Tomaszewski made the following conjecture. Given $n$ real numbers $a_{1},...,a_{n}$ with $\sum_{i=1}^{n}a_{i}^{2}=1$, then of the $2^{n}$ signed sums $\pm a_{1} \pm ... \pm a_{n}$, at least half have absolute value at most $1$. Hendriks and Van Zuijlen (2020) and Boppana (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 $0.46$.

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Vojtěch Dvořák, Peter van Hintum, Marius Tiba. 2020-05-11. Improved Bound for Tomaszewski's Problem. https://arxiv.org/abs/2005.05031

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