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Claim. With $f(N)$ the quantity of Problem 202 and $L(N)=\exp(\sqrt{\log N\log\log N})$,

$$f(N)\le NL(N)^{-1/2+o(1)},$$

with no restriction on the moduli. Croot had proved the coefficient $1/2$ for squarefree moduli and $1/6$ in general; Chen removes the squarefree restriction by counting the high-power parts of the moduli, selecting a common residue and rescaling. Y.-G. Chen, On disjoint arithmetic progressions, Acta Arith. 118 (2005), no. 2, 143–148, cited as [Ch05] on the problem page; the library's theorem page compiles the proof.

Covers. The upper bound above, which improved the coefficient $1/6$ of Croot's claim page and was itself improved to $\sqrt3/2$ by de la Bretèche, Ford and Vandehey. The paper gives no new lower bound and does not determine $f(N)$.

Depends on. Nothing in this wiki; the theorem is the paper's own.

Acceptance. Refereed: Acta Arithmetica 118 (2005), no. 2, 143–148, doi:10.4064/aa118-2-4. Not reviewed: the site labels the problem SOLVED (LEAN) and credits the answer to Ho's result, not to this paper. The page is named by the publication year; the journal record gives no day and no preprint posting is recorded.