Kra, Bryna A generalization of Furstenberg’s diophantine theorem. (English) Zbl 0921.11034 Proc. Am. Math. Soc. 127, No. 7, 1951-1956 (1999). The following extension of a well-known theorem of H. Furstenberg [Math. Systems Theory 1, 1-49 (1967; Zbl 0146.28502)] is proved. Let \(k\in \mathbb{N}\) and \(p_i,q_i\in \mathbb{N}\) with \(1< p_i< q_i\) for \(i=1,\dots, k\) and assume that \(p_1\leq p_2\leq\dots p_k\). Assume that the pairs \(p_i, q_i\) are multiplicatively independent for \(i=1,\dots, k\). Then for distinct \(\alpha_1,\dots, \alpha_k\in \mathbb{T}\) with at least one \(\alpha_i\not\in \mathbb{Q}\) \[ \Biggl\{ \sum_{i=1}^k p_i^n q_i^m \alpha_i: n,m\in \mathbb{N} \Biggr\} \] is dense in the torus \(\mathbb{T}\). Reviewer: R.F.Tichy (Graz) Cited in 4 ReviewsCited in 8 Documents MSC: 11J71 Distribution modulo one 54H20 Topological dynamics (MSC2010) Keywords:Furstenberg’s diophantine theorem; topological dynamics; distribution modulo 1; dense sets in the torus Citations:Zbl 0146.28502 PDFBibTeX XMLCite \textit{B. Kra}, Proc. Am. Math. Soc. 127, No. 7, 1951--1956 (1999; Zbl 0921.11034) Full Text: DOI