A sharp result on $m$-covers
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- by Hao Pan and Zhi-Wei Sun PDF
- Proc. Amer. Math. Soc. 135 (2007), 3515-3520 Request permission
Abstract:
Let $A=\{a_{s}+n_{s}\mathbb Z \}_{s=1}^{k}$ be a finite system of residue classes which forms an $m$-cover of $\mathbb Z$ (i.e., every integer belongs to at least $m$ members of $A$). In this paper we show the following sharp result: For any positive integers $m_{1},\ldots ,m_{k}$ and $\theta \in [0,1)$, if there is $I\subseteq \{1,\ldots ,k\}$ such that the fractional part of $\sum _{s\in I} m_{s}/n_{s}$ is $\theta$, then there are at least $2^{m}$ such subsets of $\{1,\ldots ,k\}$. This extends an earlier result of M. Z. Zhang and an extension by Z. W. Sun. Also, we generalize the above result to $m$-covers of the integral ring of any algebraic number field with a power integral basis.References
- 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, DOI 10.1007/978-3-642-60408-9_{3}
- Richard K. Guy, Unsolved problems in number theory, 3rd ed., Problem Books in Mathematics, Springer-Verlag, New York, 2004. MR 2076335, DOI 10.1007/978-0-387-26677-0
- James H. Jordan, A covering class of residues with odd moduli, Acta Arith. 13 (1967/68), 335–338. MR 220657, DOI 10.4064/aa-13-3-335-338
- 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
- Zhi Wei Sun, Covering the integers by arithmetic sequences, Acta Arith. 72 (1995), no. 2, 109–129. MR 1347259, DOI 10.4064/aa-72-2-109-129
- Zhi-Wei Sun, Covering the integers by arithmetic sequences. II, Trans. Amer. Math. Soc. 348 (1996), no. 11, 4279–4320. MR 1360231, DOI 10.1090/S0002-9947-96-01674-1
- Zhi-Wei Sun, Exact $m$-covers and the linear form $\sum ^k_{s=1}x_s/n_s$, Acta Arith. 81 (1997), no. 2, 175–198. MR 1456240, DOI 10.4064/aa-81-2-175-198
- Zhi-Wei Sun, On covering multiplicity, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1293–1300. MR 1486752, DOI 10.1090/S0002-9939-99-04817-0
- Zhi-Wei Sun, Unification of zero-sum problems, subset sums and covers of $\Bbb Z$, Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 51–60. MR 1988872, DOI 10.1090/S1079-6762-03-00111-2
- Zhi-Wei Sun, On the range of a covering function, J. Number Theory 111 (2005), no. 1, 190–196. MR 2124049, DOI 10.1016/j.jnt.2004.11.004
- Z. W. Sun, A connection between covers of the integers and unit fractions, Adv. in Appl. Math., 38 (2007), 267–274.
- 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
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
- MR Author ID: 254588
- Email: zwsun@nju.edu.cn
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2336565