• Rogers theorem addresses density of integers remaining after removing finite congruence classes in sieve theory. • The theorem appears only in Halberstam & Roth’s 1966 text, unpublished elsewhere. • It has been cited in just three papers: Lewis 1996, Filaseta et al. 2007, Ford 2008. • Rogers theorem informs the density calculation for periodic sieved sets in the cyclic group. • Recent work on Erdős problem 281 used Rogers theorem to simplify the solution. • AI‑driven proofs and Koishi Chan’s reduction highlight the theorem’s unexpected modern relevance.

Article Summaries:

  • Rogers’ theorem, first noted in a 1966 Halberstam-Roth text, gives a simple extremal result in sieve theory: for a fixed number of congruence classes removed from the integers, the remaining set’s density is maximised when all moduli are pairwise coprime. The theorem has been cited only three times in the literature and is not widely known, partly because it was unpublished and not available online. Its recent rediscovery helped resolve Erdős-Graham’s 1980 problem on sieving densities, showing that the result reduces the problem to a 1936 theorem of Davenport and Erdős. The proof relies on a “symmetrisation” or compression argument in finite cyclic groups.

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