On covering multiplicity

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corollary_2: Sun's corollary for an m-cover with moduli in increasing order: if the first k-1 classes are not an m-cover but their reciprocal moduli sum to m, then the two largest moduli are equal and exceed 1, and every multiple of 1/n_k in [0,1) is a subset sum of the other reciprocals.

corollary_4: Sun's corollary for a minimal m-cover and positive m_s prime to n_s: for each t, every r/n_t with 0 <= r < n_t is the fractional part of a difference of two subset sums of m_s/n_s over sets avoiding t, each sum at least m-1.

remark_2: Sun's remark that, for an m-cover whose largest modulus is unique, the reciprocals of the other moduli sum to at least m, and to more than m when those classes alone are not an m-cover, extending Erdős's statement that a covering with distinct moduli above 1 has reciprocal sum above 1.

theorem_1: Sun's main theorem: for a finite system of residue classes with covering multiplicity m(A), every fractional part of a subset sum of m_s/n_s recurs for at least m(A) other subsets, and a point covered exactly m(A) times forces a full coset of fractions with denominator N(J) among the fractional parts of the subset sums.

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Zhi-Wei Sun, On covering multiplicity, Proceedings of the American Mathematical Society 127 (1999), no. 5, 1293–1300, doi:10.1090/S0002-9939-99-04817-0.

For a finite system $A=\{a_s(n_s)\}_{s=1}^k$ of residue classes $a_s(n_s)=a_s+n_s\mathbb Z$ with positive moduli, Sun defines the covering multiplicity $m(A)=\inf_{x\in\mathbb Z}|\{s:x\equiv a_s\pmod{n_s}\}|$ and relates it to the fractional parts of subset sums $\sum_{s\in I}m_s/n_s$. The main result, Theorem 1 (preprint p. 2), says that each such fractional part recurs for at least $m(A)$ other subsets, and that a point covered exactly $m(A)$ times forces a full coset $\{(\alpha+r)/N(J):0\le r<N(J)\}$ among the fractional parts. Four corollaries follow; two of them constrain the moduli of $m$-covers: Corollary 2 (p. 3) when the classes other than the one of largest modulus fail to be an $m$-cover while their reciprocal moduli sum to $m$, and Corollary 4 (p. 4) for minimal $m$-covers. Remark 2 (p. 4) combines Corollary 2 with an inequality from the author's 1996 paper to extend Erdős's statement that a 1-cover with distinct moduli above 1 has reciprocal sum above 1. The proofs rest on a characterization of $m$-covers by exponential sums quoted from the author's earlier work.

The copy read for this card is the author's preprint, which has nine internally numbered pages and carries the published journal citation. It is distinguished from a publisher facsimile; result locators in this digest and on the result pages refer to its own pagination, and the labels are its own. It prints no copyright or license line; the publisher's article page (https://www.ams.org/journals/proc/1999-127-05/S0002-9939-99-04817-0/, read 2026-10-07) prints "© Copyright 1999 American Mathematical Society", so the term is reserved.

Bears on. #947: Remark 2 (p. 4) extends Erdős's statement that every 1-cover with distinct moduli above 1 has reciprocal sum above 1; with the paper's (3), by which an exact 1-cover has reciprocal sum exactly 1, that statement excludes an exact cover with distinct moduli above 1. The paper does not draw this conclusion, and the remark's first inequality is cited from the author's 1996 paper rather than proved here. #1189: a covering choice of residues on an irreducible covering set is a minimal 1-cover, so consequence (b) of Theorem 1 (p. 2) and Corollary 4 (p. 4) apply to it with $m=1$; they constrain the moduli (for instance $n_t\le2^{k-1}$ for each $t$, derived on the Theorem 1 page) but do not count irreducible covering sets, which the paper does not discuss. #1205 cites the card as context only: that problem chooses the classes and restricts the covered integers to those up to $x$, so Theorem 1 does not apply to it as stated.

Read status. Claims checked: the statements of Theorem 1, consequences (a) and (b), Corollaries 2 and 4 and Remark 2 were read clause by clause on the page images. The proofs were read but not checked step by step, and nothing here is independently reviewed.

Results. Page numbers are those of the preprint (pp. 1--9).

• Theorem 1 (p. 2; proof pp. 5--8): (i) for every $J\subseteq\{1,\ldots,k\}$ and all integers $m_1,\ldots,m_k$, at least $m(A)$ subsets $I\ne J$ give $\sum_{s\in I}m_s/n_s$ the same fractional part as $\sum_{s\in J}m_s/n_s$; (ii) if $\emptyset\ne J\subseteq S(x)$ for some $x$ with $|S(x)|=m(A)$ and each $m_s$ with $s\notin J$ is a positive integer prime to $n_s$, then for some $\alpha\in[0,1)$ every $(\alpha+r)/N(J)$, $0\le r<N(J)$, is $\{\sum_{s\in I}m_s/n_s\}$ for some $I$ disjoint from $J$ with $[\sum_{s\in I}m_s/n_s]\ge m(A)-|J|$, where $N(J)$ is the least common multiple of the $n_s$ with $s\in J$. The page also records the consequences (a) and (b) for $m$-covers stated on p. 2.
• Corollary 1 (p. 3): for an $m$-cover and any integers $m_1,\ldots,m_k$, the fractional parts $\{\sum_{s\in I}m_s/n_s\}$ take at most $2^k/(m+1)$ values.
• Corollary 2 (p. 3; proof pp. 3--4): for an $m$-cover with $n_1\le\cdots\le n_{k-1}\le n_k$ whose first $k-1$ classes are not an $m$-cover, $\sum_{s=1}^{k-1}1/n_s=m$ implies $n_{k-1}=n_k>1$ and that every $r/n_k$, $0\le r<n_k$, is a subset sum of $1/n_1,\ldots,1/n_{k-1}$.
• Remark 2 (p. 4): for an $m$-cover with $n_1\le\cdots\le n_{k-1}<n_k$, $\sum_{s=1}^{k-1}1/n_s\ge m$ (cited from the 1996 paper), with strict inequality when the first $k-1$ classes are not an $m$-cover.
• Corollary 3 (p. 4): for an $m$-cover, if $\emptyset\ne J$ and some $x$ lies in exactly $m-|J|$ classes $a_s(n_s)$ with $s\notin J$, then for any signs $\varepsilon_s\in\{1,-1\}$ the fractional parts $\{\sum_{s\in I}\varepsilon_s/n_s\}$, $I\subseteq J^-$, take at least $N(J)$ values.
• Corollary 4 (p. 4; proof p. 5): for a minimal $m$-cover and positive $m_s$ prime to $n_s$, every $r/n_t$, $0\le r<n_t$, is the fractional part of $\sum_{s\in I}m_s/n_s-\sum_{s\in J}m_s/n_s$ for some $I,J\subseteq\{1,\ldots,k\}\setminus\{t\}$ with both sums at least $m-1$.
• Lemma 1 (p. 5): for positive integers $k,m,n$ with $k>m-n\ge0$, $A$ is an $m$-cover if and only if deleting any $m-n$ of its classes leaves an $n$-cover. Lemma 2 (p. 7) is the invariance step in the proof of Theorem 1(ii).

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