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Claim. For fixed moduli $n_1, \ldots, n_r$ the density of the integers lying in none of the classes $a_i \pmod{n_i}$ is largest when every $a_i = 0$, so the minimum covered density is attained when all the residues agree: the answer to the second question of Problem 278 is yes. The theorem is Rogers', first published in H. Halberstam and K. F. Roth, Sequences, Vol. I (Clarendon Press, Oxford, 1966), pp. 242--244, reissued by Springer in 1983 (Section 5.3), cited as [HaRo66] on the problem page. Filaseta, Ford, Konyagin, Pomerance and Yu cite it from the 1966 edition as Rogers' theorem (library card), and Schroeder's manuscript (claim page) treats the second question as settled by it. Simpson's inclusion-exclusion bound of 1986, the result the site's commentary credits for the second question, states the same extremal fact with the extremal value written out (claim page); a comment of 2026-03-03 on the problem's discussion thread reports the book's earlier publication. The same theorem supplies one step of the elementary proof of Problem 281 recorded on KoishiChan's claim page, which also records a proof of the theorem posted by Terence Tao on his blog on 2026-01-19.

Covers. The second question alone: equal residues minimize the covered density. The first question, the value of the maximum covered density, is not addressed.

Depends on. Nothing in this wiki: the theorem is the book's own.

Standing. Claimed: the book is not a refereed journal publication, and the site's commentary on this problem credits Simpson, not Rogers, with settling the second question, so no acceptance evidence is listed. The book gives the year 1966 and no month, which names the page.