On multiplicative recurrence along linear patterns

Donoso, Le, Moreira, and Sun (J. Anal. Math. 149 (2023), 719–761) study sets of recurrence for actions of the multiplicative semigroup and provide some sufficient conditions for sets of the form to be sets of recurrence for such actions. A necessary condition for to be a set of multiplicative recurrence is that for every completely multiplicative function taking values on the unit circle, we have that . In this article, we fully characterize the integer quadruples which satisfy the latter property. Our result generalizes a result of Klurman and Mangerel (Math. Ann. 372 (2018), 651–697) concerning the pair , as well as some results from (Donoso, Le, Moreira, and Sun, J. Anal. Math. 149 (2023), 719–761). In addition, we prove that, under the same conditions on , the set is a set of recurrence for finitely generated actions of .