Properties of a certain sequence

Question

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.

Answer 1

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 $[0,1)$.