Dense (mod 1) dilated semigroups of algebraic numbers

Given a real algebraic number field K we consider the following possible properties of a multiplicative subsemigroup S of K^*: (1) Sα is dense modulo 1 for every α ∋ K. (2) Sα is dense modulo 1 for every α ≠ 0. A full characterization of those semigroups satisfying each of these properties is obtained. In particular, it follows that a semigroup possessing one of these properties has a subsemigroup, generated by two elements, with the same property. Given a finitely generated semigroup, one can effectively decide whether or not it satisfies either one of the aforementioned properties. A p-adic analogue of the main result is studied as well.