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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Covering the integers by arithmetic sequences. II
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by Zhi-Wei Sun PDF
Trans. Amer. Math. Soc. 348 (1996), 4279-4320 Request permission

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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Additional Information
  • 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
  • MR Author ID: 254588
  • Email: zhiwei@science.unitn.it
  • 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.
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 4279-4320
  • MSC (1991): Primary 11B25; Secondary 11A07, 11B75, 11D68
  • DOI: https://doi.org/10.1090/S0002-9947-96-01674-1
  • MathSciNet review: 1360231