Sieving by large integers and covering systems of congruences
Authors:
Michael Filaseta, Kevin Ford, Sergei Konyagin, Carl Pomerance and Gang Yu
Journal:
J. Amer. Math. Soc. 20 (2007), 495517
MSC (2000):
Primary 11B25, 11A07, 11N35
Published electronically:
September 19, 2006
MathSciNet review:
2276778
Fulltext PDF Free Access
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Abstract: An old question of Erdos asks if there exists, for each number , a finite set of integers greater than and residue classes for whose union is . We prove that if is bounded for such a covering of the integers, then the least member of is also bounded, thus confirming a conjecture of Erdos and Selfridge. We also prove a conjecture of Erdos and Graham, that, for each fixed number , the complement in of any union of residue classes , for distinct , has density at least for sufficiently large. Here is a positive number depending only on . Either of these new results implies another conjecture of Erdos and Graham, that if is a finite set of moduli greater than , with a choice for residue classes for which covers , then the largest member of cannot be . 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
Email:
filaseta@math.sc.edu
Kevin Ford
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email:
ford@math.uiuc.edu
Sergei Konyagin
Affiliation:
Department of Mathematics, Moscow State University, Moscow 119992, Russia
Email:
konyagin@ok.ru
Carl Pomerance
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, New Hamphshire 037553551
Email:
carl.pomerance@dartmouth.edu
Gang Yu
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
Email:
yu@math.kent.edu
DOI:
http://dx.doi.org/10.1090/S0894034706005492
PII:
S 08940347(06)005492
Keywords:
Covering system.
Received by editor(s):
May 25, 2005
Published electronically:
September 19, 2006
Additional Notes:
The first author was supported by NSF grant DMS0207302 and NSA grant H982300510038.
The second author was supported by NSF grant DMS0301083.
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 DMS0200187) and the University of Illinois at UrbanaChampaign in February 2004 (supported by NSF grant DMS0301083).
The fourth author was supported by NSF grant DMS0401422.
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 DMS0601033.
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
