Introduction to Zhi-Wei Sun's Papers on Covers

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generating_theorem: Generates every finite weighted residue system whose covering-map values lie in a prescribed subset of an additive commutative monoid.

main_theorem_on_equivalence: Characterizes maps whose weighted sums agree for all equivalent residue systems by a prime-refinement functional equation.

result_p7: The survey reports that the Billik–Edgar question on minimal covers with distinct moduli and prescribed greatest common divisor is equivalent to Erdős's question on covers with distinct moduli above any bound, and that under a non-existence hypothesis some modulus is divisible by three or four times the least one.

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Zhi-Wei Sun, Introduction to Zhi-Wei Sun's Papers on Covers, author survey, 22 pp., last updated 20 March 2005.

The edition cited is the 22-page version whose first page names its title, author and update date. It prints no copyright or license line, and no public address or hosting terms are recorded for it; the term is unstated.

This is a secondary author survey and bibliography. Its statements summarize the cited research papers; the survey does not replace those papers as proof sources. In particular, p. 4 identifies Finite coverings of groups as Fundamenta Mathematicae 134 (1990), no. 1, 37--53. That primary article has its own source unit and result pages.

The selected mathematical content is the survey's restatement of the Generating Theorem from Sun's 1989 paper Systems of congruences with multipliers, followed by the Main Theorem on the Equivalence, and its item 8 summary of Sun's 1991 paper On covering systems with distinct moduli. The first describes how every weighted system whose covering-map values lie in a fixed set is generated from systems of whole-line classes by prime refinements. The second characterizes functions whose weighted sums are invariant under equivalent systems. The third relates Erdős's question on covers with distinct large moduli to a question of Billik and Edgar.

Compiled scope

PDF pp. 1--3 were read for the definitions and the first two statements, p. 4 for the bibliographic identity of the 1990 group-cover paper, and p. 7 for item 8. The survey's other paper summaries and formulas are not compiled here; several restate results of primary papers that have their own source units. The original 1989 and 1991 research articles and their proofs were not compiled or independently checked.

Results restated by the survey

• The Generating Theorem (pp. 1--2)
• Main Theorem on the Equivalence (p. 2)
• Item 8: covers with distinct moduli (p. 7)

Bears on.

• Problem 2: item 8 (p. 7) states Erdős's question in the distinct-moduli form, reports its equivalence with the Billik–Edgar question, and reports that a cover with least modulus $n_1$ has some modulus divisible by $3n_1$ or $4n_1$ when no cover has distinct moduli all greater than $n_1$. Neither statement answers the question.

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