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

Published electronically:
August 15, 2007

MathSciNet review:
2336565

Full-text PDF Free Access

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]**Paul Erdős,*Some of my favorite problems and results*, The mathematics of Paul Erdős, I, Algorithms Combin., vol. 13, Springer, Berlin, 1997, pp. 47–67. MR**1425174**, 10.1007/978-3-642-60408-9_3**[G04]**Richard K. Guy,*Unsolved problems in number theory*, 3rd ed., Problem Books in Mathematics, Springer-Verlag, New York, 2004. MR**2076335****[J68]**James H. Jordan,*A covering class of residues with odd moduli*, Acta Arith.**13**(1967/1968), 335–338. MR**0220657****[PS]**Š. Porubský and J. Schönheim,*Covering systems of Paul Erdős. Past, present and future*, Paul Erdős and his mathematics, I (Budapest, 1999) Bolyai Soc. Math. Stud., vol. 11, János Bolyai Math. Soc., Budapest, 2002, pp. 581–627. MR**1954716****[S95]**Zhi Wei Sun,*Covering the integers by arithmetic sequences*, Acta Arith.**72**(1995), no. 2, 109–129. MR**1347259****[S96]**Zhi-Wei Sun,*Covering the integers by arithmetic sequences. II*, Trans. Amer. Math. Soc.**348**(1996), no. 11, 4279–4320. MR**1360231**, 10.1090/S0002-9947-96-01674-1**[S97]**Zhi-Wei Sun,*Exact 𝑚-covers and the linear form ∑^{𝑘}_{𝑠=1}𝑥_{𝑠}/𝑛_{𝑠}*, Acta Arith.**81**(1997), no. 2, 175–198. MR**1456240****[S99]**Zhi-Wei Sun,*On covering multiplicity*, Proc. Amer. Math. Soc.**127**(1999), no. 5, 1293–1300. MR**1486752**, 10.1090/S0002-9939-99-04817-0**[S03]**Zhi-Wei Sun,*Unification of zero-sum problems, subset sums and covers of ℤ*, Electron. Res. Announc. Amer. Math. Soc.**9**(2003), 51–60. MR**1988872**, 10.1090/S1079-6762-03-00111-2**[S05]**Zhi-Wei Sun,*On the range of a covering function*, J. Number Theory**111**(2005), no. 1, 190–196. MR**2124049**, 10.1016/j.jnt.2004.11.004**[S07]**Z. W. Sun,*A connection between covers of the integers and unit fractions*, Adv. in Appl. Math.,**38**(2007), 267-274.**[Z89]**Ming Zhi Zhang,*A note on covering systems of residue classes*, Sichuan Daxue Xuebao**26**(1989), no. Special Issue, 185–188 (Chinese, with English summary). MR**1059702**

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.