  Let $\vec{w} = (w_1,\dots, w_n) \in \mathbb{R}^{n}$. We show that for any
$n^{-2}\le\epsilon\le 1$, if \[\#\{\vec{\xi} \in \{0,1\}^{n}: \langle
\vec{\xi}, \vec{w} \rangle = r\} \ge 2^{-\epsilon n}\cdot 2^{n}\] for some $r
\in \mathbb{R}$, then \[\#\{\langle \vec{\xi}, \vec{w} \rangle : \vec{\xi} \in
\{0,1\}^{n}\} \le 2^{O(\sqrt{\epsilon}n)}.\] This exponentially improves a
recent result of Nederlof, Pawlewicz, Swennenhuis, and W\k{e}grzycki and leads
to a similar improvement in the parameterized (by the number of bins) runtime
of bin packing.
