## 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

### Citations:

Zbl 1164.11026; Zbl 1363.11072
Full Text:

### References:

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