Berend, Daniel Dense (mod 1) dilated semigroups of algebraic numbers. (English) Zbl 0623.10038 J. Number Theory 26, 246-256 (1987). 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. Reviewer: Harald Niederreiter (Singapore) Cited in 2 ReviewsCited in 4 Documents MSC: 11K06 General theory of distribution modulo \(1\) Keywords:dilatations; uniform distribution mod 1; real algebraic number field; multiplicative subsemigroups; p-adic analog PDFBibTeX XMLCite \textit{D. Berend}, J. Number Theory 26, 246--256 (1987; Zbl 0623.10038) Full Text: DOI References: [1] Berend, D., Multi-invariant sets on compact abelian groups, Trans. Amer. Math. Soc., 286, 505-535 (1984) · Zbl 0523.22004 [2] Berend, D., Minimal sets on tori, Ergodic Theory Dynamical Systems, 4, 499-507 (1984) · Zbl 0563.58020 [3] Furstenberg, H., Disjointness in ergodic theory, minimal sets and a problem in diophantine approximations, Math. Systems Theory, 1, 1-49 (1967) · Zbl 0146.28502 [4] Koblitz, N., \((p\)-Adic Numbers, \(p\)-adic Analysis, and Zeta-Functions (1977), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0364.12015 [5] Kuipers, L.; Niederreiter, H., (Uniform Distribution of Sequences (1974), Wiley: Wiley New York) · Zbl 0281.10001 [6] de Mathan, B., Numbers contravening a condition in density modulo 1, Acta Math. Acad. Sci. Hungar., 36, 237-241 (1980) · Zbl 0465.10040 [7] Meyer, Y., (Algebraic Numbers and Harmonic Analysis (1972), North-Holland: North-Holland Amsterdam/London) · Zbl 0267.43001 [8] Pollington, A. D., On the density of sequence {\(n_kξ\)}, Illinois J. Math., 23, 511-515 (1979) · Zbl 0413.10042 [9] Stolarsky, K. B., (Algebraic Numbers and Diophantine Approximation (1974), Dekker: Dekker New York) · Zbl 0285.10022 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.