RECORD
Anticoncentration versus the number of subset sums
Combinatorics · Submitted 19 Jan 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\vec{w} = (w_1,\dots, w_n) \in \mathbb{R}^{n}$. | background 1.00 | 0.08 | 0.01 | unclear 0.94 |
| s1 | 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. | result 0.88 | 0.95 | 0.96 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original 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.
Human review
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.