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Recurrence with prescribed number of residues. (English) Zbl 1473.11039

For a polynomial \(P(x)=x^{d}-c_{d-1}x^{d-1}-\cdots -c_{0}\in \mathbb{Z}[x]\) of degree \(d\geq 2\), an integer sequence \(S=(s_{n})_{n\in \mathbb{N}}\), defined by \[ s_{n+d}=c_{d-1}s_{n+d-1}+\cdots +c_{0}s_{n},\ \forall n\in \mathbb{N}, \] is called a linear recurrence sequence over \(\mathbb{Z}\) with characteristic polynomial \(P\). Such a sequence \(S\) is ultimately periodic modulo \(M\) for every \(M\in \mathbb{N}\), and according to [A. Dubickas, Arch. Math., Brno 42, No. 2, 151–158 (2006; Zbl 1164.11026); the reviewer et al., Arch. Math., Brno 51, No. 3, 153–161 (2015; Zbl 1363.11072)], if \(P\) is the minimal polynomial of a Pisot number \(\alpha \) and \(0\notin R:=\{s_{n}\bmod M\mid n\in \mathbb{N}\}\) for some \(M\in \mathbb{N}\), then there is \(\zeta \in \mathbb{Q}(\alpha)\) such that the sequence of the fractional parts \((\{\zeta \alpha^{n}\})_{n\in \mathbb{N}}\) has exactly \(\operatorname{card}(R)\) limit points.
In this context, the authors of the paper under review consider the case where \(P(x)= x^{2}-x-1\), i. e., \(P\) is the minimal polynomial of the Pisot number \((1+\sqrt{5})/2\). They show that for every \(r\in \mathbb{N}\) there exists \((s_{1},s_{2},M)\in \mathbb{N}^{3}\) such that the related sequence \(S=(s_{n})_{n\in \mathbb{N}}\), defined by the Fibonacci recurrence \(s_{n+2}=c_{1}s_{n+1}+c_{0}s_{n}\), \(n=1,2,3,\dots\), has exactly \(r\) distinct residues modulo \(M\), and \(0\notin R\Leftrightarrow r\geq 4\).
From this theorem, whose proof requires long verifications and non-trivial experiments with large integers, the authors obtain that for every integer \(r\geq 2\) there is \(\zeta \in \mathbb{Q}(\sqrt{5})\) such that the sequence \((\{\zeta ((1+\sqrt{5})/2)^{n}\})_{n\in \mathbb{N}}\) has exactly \(r\) limit points.

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11B50 Sequences (mod \(m\))
11J71 Distribution modulo one
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
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References:

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