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Source. Zhi-Wei Sun, Covering the integers by arithmetic sequences II, Trans. Amer. Math. Soc. 348 (1996), no. 11, 4279–4320, DOI. Theorem II is on p. 7 of the 48-page author copy described on the source card, which does not carry the journal pagination. Like Theorem I, it collects results the paper proves in more general forms later.

Conventions

The conventions are those of Theorem I: $A=\{a_s+n_s\mathbb Z\}_{s=1}^k$ with $a_s\in\mathbb Z$, $n_s\in\mathbb Z^+$; $(x,y)$ is the greatest common divisor; the denominator of a rational $a/b$ with $b\in\mathbb Z^+$ and $(a,b)=1$ is $b$. The paper's conditions (7) and (8) (p. 5) are

$$n_1\le\cdots\le n_{k-l}<n_{k-l+1}=\cdots=n_k, \qquad n_1<\cdots<n_{k-l}<n_{k-l+1}=\cdots=n_k .$$

Statement

Let $A$ be an exact $m$-cover of $\mathbb Z$ with $m\in\mathbb Z^+$.

(i) Let $n\in\mathbb Z^+$ and let $v$ be a rational such that exactly one $J\subseteq\{1,\dots,k\}$ has $\sum_{s\in J}(n,n_s)/n_s=v$ (for example $v=0$). Then for every $t=1,\dots,k$ there is $I\subseteq\{1,\dots,k\}$ with $t\in I$ and

$$\sum_{s\in I}\frac{(n,n_s)}{n_s}\equiv v\pmod 1 .$$

(ii) Assume (8) with $0<l\le k$. Then $n_s\mid n_k$ for all $s=1,\dots,k$, and for each $r\in\mathbb Z$ there is $I\subseteq\{1,\dots,k-1\}$ with

$$\sum_{s\in I}\frac{n_k}{n_s}\equiv r\pmod{n_k}$$

(the paper's (14)).

(iii) If $m=1$, then for all $t=1,\dots,k$ and $r=0,1,\dots,n_t-1$ there is $I\subseteq\{1,\dots,k\}$ with $t\notin I$ and

$$\frac r{n_t}=\sum_{s\in I}\frac1{n_s}$$

(the paper's (15)); the equality is exact, not modulo $1$.

(iv) Assume (7) with $0<l<k$. For every positive integer $\lambda<n_k/n_{k-l}$, the binomial coefficient $\binom l\lambda$ is a sum of denominators greater than $1$ of rationals

$$\sum_{s\in I}\frac1{n_s}-\frac\lambda{n_k},\qquad I\subseteq\{1,\dots,k\}.$$

The paper writes the range as $\lambda<n_k/n_{k-l}\le l$; the inequality $n_k/n_{k-l}\le l$ for an exact $m$-cover under (7) with $0<l<k$ is the consequence of Sun's earlier improvement of the Newman–Znám result recalled on p. 5.

Proof route and dependencies

As the paper's remarks state:

• (i) from Corollary 11 applied to the complementary set (remark, p. 39);
• (ii) is equivalent to Corollary 5(ii) (remark, p. 14), proved on p. 13 from Theorem 1 and an earlier theorem of Sun;
• (iii) is Corollary 9 with $n=1$ (p. 24); the paper contrasts it with a conjecture of Z. H. Sun that its Examples 1 and 2 refute (pp. 14–15);
• (iv) from Corollary 12, since $\sum_{s=1}^k1/n_s=m$ (p. 40).

No proof is reconstructed here.

Bears on

No row. The statements concern exact $m$-covers in general, and no problem page uses them. The exclusion of exact covers with distinct moduli (Problem 947) is recorded under part (iv) of Theorem I.