Geodesic
Anticoncentration versus the number of subset sums
Advances in Combinatronics (2021)

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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.
Publié le : 
@article{ADVC_2021_a3,
     author = {Vishesh Jain and Ashwin Sah and Mehtaab Sawhney},
     title = {Anticoncentration versus the number of subset sums},
     journal = {Advances in Combinatronics},
     publisher = {mathdoc},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADVC_2021_a3/}
}
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%A Ashwin Sah
%A Mehtaab Sawhney
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Vishesh Jain; Ashwin Sah; Mehtaab Sawhney. Anticoncentration versus the number of subset sums. Advances in Combinatronics (2021). http://geodesic.mathdoc.fr/item/ADVC_2021_a3/