  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 $0.46$.
