Anticoncentration versus the number of subset sums

TL;DR

This paper establishes a new bound on the number of distinct subset sums based on anticoncentration properties, improving previous results and impacting algorithms for bin packing.

Contribution

It provides an exponential improvement in the dependence on  in bounds for subset sums, advancing understanding of anticoncentration and its algorithmic applications.

Findings

Bound on the number of subset sums with large concentration

Improved  dependence in subset sum bounds

Enhanced parameterized algorithms for bin packing

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 \[\#\{\vec{\xi} \in \{0,1\}^{n}: \langle \vec{\xi}, \vec{w} \rangle = \tau\} \ge 2^{-\epsilon n}\cdot 2^{n}\] 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 W\k{e}grzycki and leads to a similar improvement in the parameterized (by the number of bins) runtime of bin packing.