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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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Sieving by large integers and covering systems of congruences
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by Michael Filaseta, Kevin Ford, Sergei Konyagin, Carl Pomerance and Gang Yu PDF
J. Amer. Math. Soc. 20 (2007), 495-517 Request permission

Abstract:

An old question of Erdős asks if there exists, for each number $N$, a finite set $S$ of integers greater than $N$ and residue classes $r(n)~(\textrm {mod}~n)$ for $n\in S$ whose union is $\mathbb Z$. We prove that if $\sum _{n\in S}1/n$ is bounded for such a covering of the integers, then the least member of $S$ is also bounded, thus confirming a conjecture of Erdős and Selfridge. We also prove a conjecture of Erdős and Graham, that, for each fixed number $K>1$, the complement in $\mathbb Z$ of any union of residue classes $r(n)~(\textrm {mod}~n)$, for distinct $n\in (N,KN]$, has density at least $d_K$ for $N$ sufficiently large. Here $d_K$ is a positive number depending only on $K$. Either of these new results implies another conjecture of Erdős and Graham, that if $S$ is a finite set of moduli greater than $N$, with a choice for residue classes $r(n)~(\textrm {mod}~n)$ for $n\in S$ which covers $\mathbb Z$, then the largest member of $S$ cannot be $O(N)$. We further obtain stronger forms of these results and establish other information, including an improvement of a related theorem of Haight.
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Additional Information
  • Michael Filaseta
  • Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
  • MR Author ID: 66800
  • Email: filaseta@math.sc.edu
  • Kevin Ford
  • Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
  • MR Author ID: 325647
  • ORCID: 0000-0001-9650-725X
  • Email: ford@math.uiuc.edu
  • Sergei Konyagin
  • Affiliation: Department of Mathematics, Moscow State University, Moscow 119992, Russia
  • MR Author ID: 188475
  • Email: konyagin@ok.ru
  • Carl Pomerance
  • Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hamphshire 03755-3551
  • MR Author ID: 140915
  • Email: carl.pomerance@dartmouth.edu
  • Gang Yu
  • Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
  • Email: yu@math.kent.edu
  • Received by editor(s): May 25, 2005
  • Published electronically: September 19, 2006
  • Additional Notes: The first author was supported by NSF grant DMS-0207302 and NSA grant H98230-05-1-0038.
    The second author was supported by NSF grant DMS-0301083.
    Much of the research for this paper was accomplished while the third author was visiting the University of South Carolina, Columbia, in January 2004 (supported by NSF grant DMS-0200187) and the University of Illinois at Urbana-Champaign in February 2004 (supported by NSF grant DMS-0301083).
    The fourth author was supported by NSF grant DMS-0401422.
    The work of the last author was completed while he was at the University of South Carolina; he was supported in part by NSF grant DMS-0601033.
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 20 (2007), 495-517
  • MSC (2000): Primary 11B25, 11A07, 11N35
  • DOI: https://doi.org/10.1090/S0894-0347-06-00549-2
  • MathSciNet review: 2276778