During research I came to the following sequence:


Let $\lambda>1$
and define $n\_{k+1}=\text{IntergerPart}[\lambda\cdot n\_k]$ where we assume that $n\_0$ is sufficently large integer, so that the sequence $n\_k$ is strictly increasing. Finally let $x\_k=\text{FractionalPart}[\lambda\cdot n\_k]$.


**Question:** Are the following claims true or false?


*Claim 1:* If $\lambda\in \mathbb{R}\backslash\mathbb{Q}$
then the sequence $(x\_k)\_k$ is dense in $[0,1)$.


*Claim 2:* If $\lambda\in \mathbb{Q}\backslash\mathbb{N}$
then the sequence $(x\_k)\_k$ is not periodic.


Since I'm not a specialist in this field I'm looking for some hints how to prove or tackle these two problems. Claims were made after some computer experiments.


P.s. It is clear that in Claim 2 the sequence is generated by finitely many numbers.



Properties of a certain sequence
Claim 1 is false. Let $\phi = \frac{1+\sqrt{5}}{2}$, $\overline{\phi} = \frac{1-\sqrt{5}}{2}$, $\lambda = \phi^{2}$ and $n\_{0} = 1$.


I claim that $n\_{k} = F\_{2k+1}$, the $(2k+1)$st Fibonacci number for $k \geq 1$.
By strong induction and Binet's formula $F\_{k} = \frac{1}{\sqrt{5}}\left(\phi^{k} - \overline{\phi}^{k}\right)$, we have
$$
 n\_{k} = \lfloor \phi^{2} F\_{2k-1} \rfloor = \left\lfloor \phi^{2} \cdot \frac{1}{\sqrt{5}} (\phi^{2k-1} - \overline{\phi}^{2k-1}) \right\rfloor = \left\lfloor F\_{2k+1} - \frac{1}{\sqrt{5}} \overline{\phi}^{2k-3} (1-\overline{\phi}^{2})\right\rfloor.
$$
Since $-1 < \overline{\phi} < 0$, the second term in the expression above is positive and less than $1$. Thus, $n\_{k} = F\_{2k+1}$.


Hence
$$
x\_{k} = \phi^{2} F\_{2k-1} - n\_{k} = -\frac{1}{\sqrt{5}} \overline{\phi}^{2k-3}(1-\overline{\phi}^{2})
$$
which tends to $0$ exponentially as $k \to \infty$. In particular, $( x\_{k} )\_{k}$ is not dense in $[0,1)$.


