Anticoncentration versus the number of subset sums

Abstract

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

10 pages; revised version