Addendum To: a Theorem of Khintchine Type

In this addendum to a previous Khintchine-type theorem, the author extends a metric Diophantine approximation result concerning the Lebesgue measure of sets of real numbers approximable by infinitely many reduced rationals. Assuming suitable divergence and regularity conditions on the approximation function, it is proved that the full-measure conclusion remains valid when the denominators are restricted to a strictly increasing sequence of natural numbers with positive lower density. The proof combines elementary methods from measure theory and arithmetic with Gallagher’s zero-one law.