A sharp result on -covers

Authors:
Hao Pan and Zhi-Wei Sun

Journal:
Proc. Amer. Math. Soc. **135** (2007), 3515-3520

MSC (2000):
Primary 11B25; Secondary 11B75, 11D68, 11R04

DOI:
https://doi.org/10.1090/S0002-9939-07-08890-9

Published electronically:
August 15, 2007

MathSciNet review:
2336565

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a finite system of residue classes which forms an -cover of (i.e., every integer belongs to at least members of ). In this paper we show the following sharp result: For any positive integers and , if there is such that the fractional part of is , then there are at least such subsets of . This extends an earlier result of M. Z. Zhang and an extension by Z. W. Sun. Also, we generalize the above result to -covers of the integral ring of any algebraic number field with a power integral basis.

**[E97]**P. Erdos,*Some of my favorite problems and results,*in: The Mathematics of Paul Erdos, I, 47-67, Algorithms Combin., 13, Springer, Berlin, 1997. MR**1425174 (98e:11002)****[G04]**R. K. Guy,*Unsolved Problems in Number Theory*, 3rd edition, Springer, New York, 2004. MR**2076335 (2005h:11003)****[J68]**J. H. Jordan,*A covering class of residues with odd moduli*, Acta Arith.**13**(1968), 335-338. MR**0220657 (36:3709)****[PS]**Š. Porubský and J. Schönheim,*Covering systems of Paul Erdös: Past, present and future*, in: Paul Erdös and his Mathematics. I (edited by G. Halász, L. Lovász, M. Simonvits, V. T. Sós), Bolyai Soc. Math. Studies 11, Budapest, 2002, pp. 581-627. MR**1954716 (2004d:11006)****[S95]**Z. W. Sun,*Covering the integers by arithmetic sequences*, Acta Arith.**72**(1995), 109-129. MR**1347259 (96k:11013)****[S96]**Z. W. Sun,*Covering the integers by arithmetic sequences.II*, Trans. Amer. Math. Soc.**348**(1996), 4279-4320. MR**1360231 (97c:11011)****[S97]**Z. W. Sun,*Exact -covers and the linear form*, Acta Arith.**81**(1997), 175-198. MR**1456240 (98h:11019)****[S99]**Z. W. Sun,*On covering multiplicity*, Proc. Amer. Math. Soc.**127**(1999), 1293-1300. MR**1486752 (99h:11012)****[S03]**Z. W. Sun,*Unification of zero-sum problems, subset sums and covers of*, Electron. Res. Announc. Amer. Math. Soc.**9**(2003), 51-60. MR**1988872 (2004i:11017)****[S05]**Z. W. Sun,*On the range of a covering function*, J. Number Theory**111**(2005), 190-196. MR**2124049 (2005m:11015)****[S07]**Z. W. Sun,*A connection between covers of the integers and unit fractions*, Adv. in Appl. Math.,**38**(2007), 267-274.**[Z89]**M. Z. Zhang,*A note on covering systems of residue classes*, Sichuan Daxue Xuebao (Nat. Sci. Ed.)**26**(1989), Special Issue, 185-188. MR**1059702 (92c:11003)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
11B25,
11B75,
11D68,
11R04

Retrieve articles in all journals with MSC (2000): 11B25, 11B75, 11D68, 11R04

Additional Information

**Hao Pan**

Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

Email:
haopan79@yahoo.com.cn

**Zhi-Wei Sun**

Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

Email:
zwsun@nju.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-07-08890-9

Received by editor(s):
January 3, 2006

Received by editor(s) in revised form:
June 3, 2006, and August 25, 2006

Published electronically:
August 15, 2007

Additional Notes:
The second author is responsible for communications and is supported by the National Science Fund for Distinguished Young Scholars (No. 10425103) in China.

Communicated by:
Wen-Ching Winnie Li

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.