The moving-target Khintchine conjecture holds under an extra-divergence condition on psi and whenever the target centers lie in a finite set, yielding a monochromatic-denominator corollary.
A note on the Duffin–Schaeffer conjecture with slow divergence.Bulletin of the London Mathematical Society, 46(1):164–168, 2014
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Toward Khintchine's theorem with a moving target: extra divergence or finitely centered target
The moving-target Khintchine conjecture holds under an extra-divergence condition on psi and whenever the target centers lie in a finite set, yielding a monochromatic-denominator corollary.