SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2239-2249, January 2020.
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.

中文翻译：

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Tomaszewski问题的改进边界

SIAM离散数学杂志，第34卷，第4期，第2239-2249页，2020
年1月。1986年，Tomaszewski做出了以下推测。给定$ n $个实数$ a_ {1}，\ ldots，a_ {n} $和$ \ sum_ {i = 1} ^ {n} a_ {i} ^ {2} = 1 $，然后是$ 2 ^ {n} $个签名的总和$ \ pm a_ {1} \ pm \ cdots \ pm a_ {n} $，至少一半具有绝对值，最高为1。Hendriks和van Zuijlen [对Rademacher的Boppana-Holzman绑定的改进随机变量}，arXiv：2003.02588，2020年）和Boppana，Hendriks和van Zuijlen [Tomaszewski的随机签名和的问题，已复习，arXiv：2003.06433，2020年]独立证明，这些和中至少有0.4276的一部分具有绝对值大多数1.使用不同的技术，我们将此边界提高到0.46。