Let $a_1, \ldots, a_n \in \mathbb{R}$ satisfy $\sum_i a_i^2 = 1$, and let $\varepsilon_1, \ldots, \varepsilon_n$ be uniformly random $\pm 1$ signs and $X = \sum_{i=1}^{n} a_i \varepsilon_i$. It is conjectured that $X = \sum_{i=1}^{n} a_i \varepsilon_i$ has $\Pr[X \geq 1] \geq 7/64$. The best lower bound so far is $1/20$, due to Oleszkiewicz. In this paper we improve this to $\Pr[X \geq 1] \geq 6/64$.

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.