Anticoncentration versus the number of subset sums

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