Dense (mod 1) dilated semigroups of algebraic numbers (Q1091425)

For a real algebraic number field \(K\) the multiplicative subsemigroups \(S\) of \(K^*\) with the property that \(S\alpha\) is dense mod 1 for every real \(\alpha\not\in K\) (or that \(S\alpha\) is dense mod 1 for every real \(\alpha\neq 0)\) are characterized. These characterizations imply, in particular, that a semigroup possessing one of these properties has a subsemigroup, generated by two elements, with the same property. For a finitely generated semigroup one can effectively decide whether or not it satisfies one of the properties given above. A \(p\)-adic analog of the characterization theorem is also established.