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On density modulo 1 of some expressions containing algebraic integers. (English) Zbl 1118.11034

H. Furstenberg [Math. Syst. Theory 1, 1–49 (1967; Zbl 0146.28502)] proved that if \(p,q>1\) are integers not both integer powers of the same integer (i.e. \(p\) and \(q\) are multiplicatively independent), then for every irrational \(\alpha\) the sequence \(p^mq^n\alpha\bmod1\), \(m,n=1,2,\dots\) is everywhere dense in \([0,1]\). B. Kra [Proc. Am. Math. Soc. 127, No. 7, 1951–1956 (1999; Zbl 0921.11034)] extended this to the following: Let \(p_i\) and \(q_i\) be integers and \(\alpha_i\) real, \(i=1,2,\dots,k\). If \(p_1,q_1>1\) are multiplicatively independent, \(\alpha_1\) is irrational, and \((p_i,q_i)\neq(p_1,q_1)\) for \(i>1\), then the sequence \(\sum_{i=1}^kp_i^mq_i^n\alpha_i\bmod1\), \(m,n=1,2,\dots\) is dense in \([0,1]\).
In this paper the author conjectures that if \(\lambda_i,\mu_i\), for \(i=1,2,\dots,k\) are real algebraic number, \(| \lambda_i| ,| \mu_i| >1\), \(\lambda_i,\mu_i\) are multiplicatively independent, and \((\lambda_i,\mu_i)\neq(\lambda_j,\mu_j)\) for \(i\neq j\), then for any real numbers \(\alpha_1,\dots,\alpha_k\) with at least one \(\alpha_i\not\in\mathbb Q(\bigcup_{i=1}^k\{\lambda_i,\mu_i\})\) the sequence \(\sum_{i=1}^k\lambda_i^m\mu_i^n\alpha_i\bmod1\), \(m,n=1,2,\dots\) is dense in \([0,1]\). He proves this for special algebraic integers of degree \(2\) and \(k=2\). As illustrating examples he give:s For any \(\alpha_1,\alpha_2\) with at least one is non-zero, the sequence \(\{(\sqrt{23}+1)^m(\sqrt{23}+2)^n\alpha_1+ (\sqrt{61}+1)^m(\sqrt{61}-6)^n\alpha_2\}, \) \(m,n=1,2,\dots\) is everywhere dense and also for irrational \(\alpha_2\) the sequence \(\{(3+\sqrt{3})^m2^n+5^m7^n\alpha_2\sqrt{2}\}\), \(m,n=1,2,\dots\), is everywhere dense in \([0,1]\). Proofs involve a consideration of some dynamical systems.

MSC:

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