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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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New bounds on the density of lattice coverings
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by Or Ordentlich, Oded Regev and Barak Weiss HTML | PDF
J. Amer. Math. Soc. 35 (2022), 295-308 Request permission

Abstract:

We obtain new upper bounds on the minimal density $\Theta _{n, \mathcal {K}}$ of lattice coverings of ${\mathbb {R}}^n$ by dilates of a convex body $\mathcal {K}$. We also obtain bounds on the probability (with respect to the natural Haar-Siegel measure on the space of lattices) that a randomly chosen lattice $L$ satisfies $L+\mathcal {K}= {\mathbb {R}}^n$. As a step in the proof, we utilize and strengthen results on the discrete Kakeya problem.
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Additional Information
  • Or Ordentlich
  • Affiliation: School of Computer Science and Engineering, Hebrew University of Jerusalem, Jerusalem 91905, Israel
  • MR Author ID: 990513
  • ORCID: 0000-0002-5791-7923
  • Oded Regev
  • Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012
  • MR Author ID: 146145
  • ORCID: 0000-0002-8616-3163
  • Barak Weiss
  • Affiliation: School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
  • MR Author ID: 335552
  • ORCID: 0000-0002-9296-3343
  • Received by editor(s): June 11, 2020
  • Received by editor(s) in revised form: April 8, 2021
  • Published electronically: July 28, 2021
  • Additional Notes: The authors were supported by grants ISF 2919/19, ISF 1791/17, BSF 2016256, the Simons Collaboration on Algorithms and Geometry, a Simons Investigator Award, and by the National Science Foundation (NSF) under Grant No. CCF-1814524.
  • © Copyright 2021 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 35 (2022), 295-308
  • MSC (2020): Primary 11H31, 94B75, 11T30
  • DOI: https://doi.org/10.1090/jams/984
  • MathSciNet review: 4322394