A sharp result on 𝑚-covers

Pan, Hao; Sun, Zhi-Wei

doi:10.1090/S0002-9939-07-08890-9

A sharp result on $m$-covers
HTML articles powered by AMS MathViewer

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