Covering the integers by arithmetic sequences II

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theorem_1: Covering as many consecutive terms of a+nZ as there are fractional parts of the subset sums of m_s/n_s, each at least m times, makes the system an m-cover of a+nZ.

theorem_i: Collects the paper's central consequences for an m-cover of the integers: integer subset sums of m_s/n_s, fractional parts when a sequence is essential, and a lower bound on how often the largest modulus repeats.

theorem_ii: Collects the paper's central consequences for an exact m-cover of the integers: residues of subset sums through a prescribed index, residues modulo the largest modulus, and binomial coefficients as sums of denominators.

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Zhi-Wei Sun, Covering the integers by arithmetic sequences II, Transactions of the American Mathematical Society 348 (1996), no. 11, 4279–4320, DOI.

This digest records the paper's place in Sun's program, its main results, and the context its introduction gives for Problem 7. Theorems I and II (pp. 6–7 of the copy read) collect the paper's central results for m-covers and exact m-covers in their simplest forms; Theorem 1 (pp. 10–11) is the finite-block criterion behind several of them. Each has a result page stating it and pointing to the paper's proof; no proof has been reconstructed or independently reviewed here. The read status of the three statements is claims checked: their hypotheses and conclusions were read clause by clause against the copy.

The copy read for this card is the author's 48-page TeX copy, internally numbered 1–48. Its first-page header carries the published Transactions citation 348 (1996), no. 11, 4279–4320; the copy itself does not use that 42-page journal pagination. Its PDF metadata identifies a TeX/pdfTeX production, so the digest cites this copy without describing it as a publisher scan or a publisher facsimile. It prints no copyright or license line, and no hosting terms are recorded for it; the term is unstated.

The paper explicitly continues the program of Sun's 1995 paper, and its introduction cites Simpson's regular-covering lower bound from 1985. Its discussion of the Erdős–Selfridge and Schinzel conjectures gives context for Problem 7; none of the statements recorded here resolves the distinct-odd-moduli question.

Results

• Theorem I (pp. 6–7): for an $m$-cover, at least $m$ positive integers are subset sums of $m_s/n_s$; integer subset sums avoiding a given index when $m>1$; fractional parts of reciprocal subset sums when a sequence is essential; and, under the paper's condition (7), $l\ge n_k/n_{k-l}$ or $\sum_{s=1}^{k-l}1/n_s\ge m$.
• Theorem II (p. 7): for an exact $m$-cover, residues of subset sums of $(n,n_s)/n_s$ and $n_k/n_s$, every $r/n_t$ as an exact subset sum of $1/n_s$ with $s\ne t$ when $m=1$, and binomial coefficients $\binom l\lambda$ as sums of denominators.
• Theorem 1 (pp. 10–11): when $(m_s,n_s)=(n,n_s)$ for every $s$, covering at least $m$ times each of as many consecutive terms of $a+n\mathbb Z$ as there are fractional parts of the sums of $m_s/n_s$ over subsets of the indices with $(n,n_s)\mid a_s-a$ forces an $m$-cover of $a+n\mathbb Z$; its part (ii) bounds the number of such fractional parts when a sequence is essential. Its proof uses Lemma 3, quoted from Sun's 1995 paper.

The paper's Theorems 2 (p. 17) and 3 (pp. 31–33) are technical general forms from which the corollaries behind Theorems I and II follow; they have no result page. Section 6 (pp. 45–46) poses eight open problems.

Bears on

• Problem 275: Theorem 1(i) with $n=1$, $m=1$ and every $m_s=1$ gives the problem's statement, since the number of fractional parts is at most $2^k$. This derivation is the corpus's; the paper credits that statement to Crittenden and Vanden Eynden and presents its Lemma 3, quoted from Sun's 1995 paper, as a stronger result (p. 10).
• Problem 947: the first assertion of Theorem I(iv) with $m=1$ excludes an exact cover by at least two sequences with distinct moduli, by the derivation on the result page; the paper credits that theorem to Davenport, Mirsky, Newman and Radó.
• Problem 7: context only; the introduction states the Erdős–Selfridge conjecture as its Conjecture II (p. 2), and no result of the paper decides it.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.