Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Covering the integers by arithmetic sequences. II

Author: Zhi-Wei Sun
Journal: Trans. Amer. Math. Soc. 348 (1996), 4279-4320
MSC (1991): Primary 11B25; Secondary 11A07, 11B75, 11D68
MathSciNet review: 1360231
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Abstract: Let $A= \{a_{s}+n_{s}\mathbb {Z}\}^{k}_{s=1}$ ($n_{1} \leqslant \cdots \leqslant n_{k})$ be a system of arithmetic sequences where $a_{1}, \cdots ,a_{k}\in \mathbb {Z}$ and $ n_{1},\cdots ,n_{k}\in \mathbb {Z}^{+}$. For $m\in \mathbb {Z}^{+}$ system $A$ will be called an (exact) $ m$-cover of $\mathbb {Z}$ if every integer is covered by $A$ at least (exactly) $m$ times. In this paper we reveal further connections between the common differences in an (exact) $ m$-cover of $\mathbb {Z}$ and Egyptian fractions. Here are some typical results for those $m$-covers $A$ of $ \mathbb {Z}$: (a) For any $ m_{1},\cdots ,m_{k}\in \mathbb {Z}^{+}$ there are at least $m$ positive integers in the form $ \Sigma _{s\in I} m_{s}/n_{s}$ where $I \subseteq \{1,\cdots ,k\}$. (b) When $n_{k-l}<n_{k-l+1}= \cdots =n_{k}$ ($0<l<k)$, either $l \geqslant n_{k}/n_{k-l}$ or $\Sigma ^{k-l}_{s=1}1/n_{s} \geqslant m$, and for each positive integer $ \lambda <n_{k}/n_{k-l}$ the binomial coefficient $ \binom l{ \lambda }$ can be written as the sum of some denominators $>1$ of the rationals $ \Sigma _{s\in I}1/n_{s}- \lambda /n_{k},\ I \subseteq \{1,\cdots ,k\}$ if $A$ forms an exact $ m$-cover of $\mathbb {Z}$. (c) If $ \{a_{s}+n_{s}\mathbb {Z}\}^{k}_{\substack {s=1\\ s\not =t}}$ is not an $ m$-cover of $\mathbb {Z}$, then $ \Sigma _{s\in I}1/n_{s},\ I \subseteq \{1,\cdots ,k\}\setminus \{t\}$ have at least $n_{t}$ distinct fractional parts and for each $ r=0,1,\cdots ,n_{t}-1$ there exist $I_{1},I_{2} \subseteq \{1,\cdots ,k\}\setminus \{t\}$ such that $ r/n_{t} \equiv \Sigma _{s\in I_{1}}1/n_{s}-\Sigma _{s\in I_{2}}1/n_{s}$ (mod 1). If $A$ forms an exact $ m$-cover of $\mathbb {Z}$ with $m=1$ or $n_{1}< \cdots <n_{k-l}<n_{k-l+1}= \cdots =n_{k}$ ($l>0$) then for every $t=1, \cdots ,k$ and $ r=0,1,\cdots ,n_{t}-1$ there is an $I \subseteq \{1,\cdots ,k\}$ such that $ \Sigma _{s\in I}1/n_{s} \equiv r/n_{t}$ (mod 1).

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Zhi-Wei Sun
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China; Dipartimento di Matematica, Università degli Studi di Trento, I-38050 Povo (Trento), Italy

Received by editor(s): June 7, 1994
Received by editor(s) in revised form: November 10, 1995
Additional Notes: This research is supported by the National Natural Science Foundation of P. R. China.
Article copyright: © Copyright 1996 American Mathematical Society