Multiplicative recurrence of M\'obius transformations

We provide a necessary and sufficient condition under which the set defined by a M\"obius transformation of positive integers is a set of multiplicative recurrence. A Khinchin-type lower bound is also established. This not only answers a question of Donoso--Le--Moreira--Sun but also strengthens their result. The key input in our proof is Tao's logarithmically averaged correlations of non-pretentious multiplicative functions.
Comment: In Lemma 4.1, [FH16, Theorem 1.4] is not applicable. In Lemma 6.2, it is not necessarily true that max{k:a^k|r} = max{k:a^k|s}