Covering the integers by arithmetic sequences. II

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).