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A generalization of Furstenberg’s diophantine theorem. (English) Zbl 0921.11034

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)

MSC:

11J71 Distribution modulo one
54H20 Topological dynamics (MSC2010)

Citations:

Zbl 0146.28502
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