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Setting

For an integer $a$ and a positive integer $n$, $a(n)=a+n\mathbb Z$. The system is

$$A=\{a_s(n_s)\}_{s=1}^k$$

with positive moduli $n_1,\ldots,n_k$ (the paper's (1), preprint p. 1). For $x\in\mathbb Z$ let $S(x)=\{1\le s\le k:x\equiv a_s\pmod{n_s}\}$; the covering multiplicity is $m(A)=\inf_{x\in\mathbb Z}\lvert S(x)\rvert$ (the paper's (2)), and $A$ is an $m$-cover when $m(A)\ge m$. A minimal $m$-cover is an $m$-cover none of whose proper subsystems is one, and an exact $m$-cover has $\lvert S(x)\rvert=m$ for every $x$ (pp. 1--2). $[x]$ and $\{x\}$ are the integral and fractional parts of a real $x$. For $J\subseteq\{1,\ldots,k\}$, $J^-=\{1,\ldots,k\}\setminus J$, and $N(J)$ is the least common multiple of the $n_s$ with $s\in J$.

The paper also records (its (3), p. 1, with a pointer to earlier work) that $\sum_{s=1}^k1/n_s\ge m(A)$, with equality if and only if $A$ covers each integer exactly $m$ times for some positive integer $m$.

Statement

Theorem 1 (preprint p. 2). Let $A$ be a system as above and $J\subseteq\{1,\ldots,k\}$.

(i) For all integers $m_1,\ldots,m_k$,

$$\Bigl\lvert\Bigl\{I\subseteq\{1,\ldots,k\}:\ I\ne J,\ \Bigl\{\sum_{s\in I}\frac{m_s}{n_s}\Bigr\} =\Bigl\{\sum_{s\in J}\frac{m_s}{n_s}\Bigr\}\Bigr\}\Bigr\rvert\ \ge\ m(A).$$

(ii) Suppose $\emptyset\ne J\subseteq S(x)$ for some $x\in\mathbb Z$ with $\lvert S(x)\rvert=m(A)$, and for each $s\in J^-$ let $m_s$ be a positive integer prime to $n_s$. Then there is $\alpha\in[0,1)$ such that for every integer $r$ with $0\le r<N(J)$ some $I\subseteq J^-$ satisfies

$$\Bigl[\sum_{s\in I}\frac{m_s}{n_s}\Bigr]\ge m(A)-\lvert J\rvert \qquad\text{and}\qquad \Bigl\{\sum_{s\in I}\frac{m_s}{n_s}\Bigr\}=\frac{\alpha+r}{N(J)}.$$

The paper writes (ii) as an inclusion of sets (its (5)): the set of fractional parts $\{\sum_{s\in I}m_s/n_s\}$ over $I\subseteq J^-$ with $[\sum_{s\in I}m_s/n_s]\ge m(A)-\lvert J\rvert$ contains $\{a/N(J):0\le a<N(J),\ \{a\}=\alpha\}$, where $a$ ranges over reals; those $a$ are exactly $\alpha+r$ with $r=0,\ldots,N(J)-1$.

Consequences stated on p. 2. The paper notes, as properties of an $m$-cover $A$ that follow from Theorem 1 (taking every $m_s=1$):

• (a) for each $J\subseteq\{1,\ldots,k\}$ there are at least $m$ subsets $I\ne J$ with $\sum_{s\in I}1/n_s-\sum_{s\in J}1/n_s\in\mathbb Z$;
• (b) if $A$ is a minimal $m$-cover, then for each $t=1,\ldots,k$ there is $\alpha_t\in[0,1)$ such that for every $r=0,1,\ldots,n_t-1$ some $I\subseteq\{1,\ldots,k\}\setminus\{t\}$ has $[\sum_{s\in I}1/n_s]\ge m-1$ and $\{\sum_{s\in I}1/n_s\}=(\alpha_t+r)/n_t$.

Property (b) is (ii) with $J=\{t\}$ (a reading of this page): since $A\setminus\{a_t(n_t)\}$ is not an $m$-cover, some $x\in a_t(n_t)$ lies in exactly $m$ classes of $A$, so $m(A)=m$ and $\{t\}\subseteq S(x)$ with $\lvert S(x)\rvert=m(A)$. The paper's Corollary 4 (p. 4) applies (ii) with $J=\{t\}$ in the same way, for positive $m_s$ prime to $n_s$, and states the result with differences of two subset sums. The paper also notes (p. 2) that the case $J=\emptyset$ of (i) for a 1-cover gives a nonempty $I$ with $\sum_{s\in I}1/n_s\in\mathbb Z$, which it attributes to M. Z. Zhang.

Source. Zhi-Wei Sun, On covering multiplicity, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1293--1300, doi:10.1090/S0002-9939-99-04817-0, read in the author's preprint identified on the source card, whose pages are numbered 1 to 9: the theorem on p. 2, the proof of (i) on pp. 5--7 and of (ii) on pp. 7--8.

Read depth. Claims checked: the definitions, the statement and the consequences (a) and (b) were read clause by clause on the page images. The proof was read but not checked step by step; nothing here is independently reviewed.

Proof pointer

Section 2 (pp. 5--8). Both parts rest on a characterization of $m$-covers that the paper quotes from the author's earlier work (Proposition 1, p. 5): a system of real arithmetic sequences $\alpha_s+\beta_s\mathbb Z$ is an $m$-cover exactly when certain signed sums of binomial coefficients times exponentials, taken over the subsets $I$ with a given fractional part $\{\sum_{s\in I}1/\beta_s\}$, vanish. Lemma 1 (p. 5) says $A$ is an $m$-cover if and only if every subsystem obtained by deleting $m-n$ classes is an $n$-cover. For (i) (pp. 5--7), applied to the rescaled classes $a_s+(n_s/m_s)\mathbb Z$, these give, case by case on whether $J$ or $J^-$ has at least $m$ elements and then on the classes of modulus 1, $m+1$ distinct subsets with the same fractional part. For (ii) (pp. 7--8), Lemma 2 (p. 7) shows that a weighted sum $C(a)$ over the subsets with fractional part $a/N(J)$ depends only on $\{a\}$; the hypothesis that $x$ lies in exactly $m(A)-\lvert J\rvert$ classes of $J^-$ and the coprimality of $m_s$ and $n_s$ make the rescaled system fail to be an $(m(A)-\lvert J\rvert+1)$-cover, so Proposition 1 yields a $\theta$ with $C(N(J)\theta)\ne0$, and $\alpha=\{N(J)\theta\}$ works.

Bears on

• Problem 1189: the problem counts irreducible covering sets, sets of distinct moduli $1<n_1<\cdots<n_k$ that admit a covering choice of residues while no proper subset does. For such a set, any covering choice of residues is a minimal 1-cover, so consequence (b) applies to it with $m=1$, where the condition $[\sum_{s\in I}1/n_s]\ge0$ is empty. The $n_t$ values $(\alpha_t+r)/n_t$ are then fractional parts of subset sums over distinct sets $I\subseteq\{1,\ldots,k\}\setminus\{t\}$, which gives $n_t\le2^{k-1}$ for every $t$ (an observation of this page; the paper draws no conclusion about irreducible covering sets). This is a necessary condition on the moduli, not a count, and the theorem does not answer the problem.