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Covering the Integers by Arithmetic Sequences. II

Zhi-Wei Sun
Transactions of the American Mathematical Society
Vol. 348, No. 11 (Nov., 1996), pp. 4279-4320
Stable URL: http://www.jstor.org/stable/2155420
Page Count: 42
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Covering the Integers by Arithmetic Sequences. II
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Abstract

Let A = {as + nsZ}k s = 1 (n1 ⩽ ⋯ ⩽ nk) be a system of arithmetic sequences where a1, ⋯, ak ∈ Z and n1, ⋯, nk ∈ Z+. For m ∈ Z+ system A will be called an (exact) m-cover of 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 Z and Egyptian fractions. Here are some typical results for those m-covers A of Z: (a) For any m1, ⋯, mk ∈ Z+ there are at least m positive integers in the form ∑s ∈ Ims/ns where $I \subseteq \{1, \cdots, k\}$. (b) When $n_{k-1} < n_{k - 1 + 1} = \cdots = n_k (0 < l < k)$, either l ⩾ nk/nk - 1 or ∑k - l s = 1 1/ns ⩾ m, and for each positive integer $\lambda < n_k/n_{k - 1}$ the binomial coefficient $\binom{l}{\lambda}$ can be written as the sum of some denominators $> 1$ of the rationals $\sum_{s \in I}1/n_s - \lambda/n_k, I \subseteq \{1, \cdots, k\}$ if A forms an exact m-cover of Z. (c) If $\{a_s + n_s\mathbb{Z}\}^k_{\substack{s = 1\\ s \neq t}}$ is not an m-cover of Z, then $\sum_{s \in I}1/n_s, I \subseteq \{1, \cdots, k\} \backslash \{t\}$ have at least nt distinct fractional parts and for each r = 0, 1, ⋯, nt - 1 there exist $I_1, I_2 \subseteq \{1, \cdots, k\} \backslash \{t\}$ such that $r/n_t \equiv \sum_{s \in I_1} 1/n_s - \sum_{s \in I_2}1/n_s (\mod 1)$. If A forms an exact m-cover of Z with m = 1 or $n_1 < \cdots < n_{k - 1} < n_{k - l + 1} = \cdots = n_k (l > 0)$ then for every t = 1, ⋯, k and r = 0, 1, ⋯, nt - 1 there is an $I \subseteq \{1, \cdots, k\}$ such that $\sum_{s \in I}1/n_s \equiv r/n_t (\mod 1)$.

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