On the Sidon tails of $\left\{\lfloor x^n\rfloor\right\}$

Abstract: We prove that the tail of the sets $$\mathbf S_x := \big{\left\lfloor x^n\right\rfloor : n\in \mathbb N\big}$$ are Sidon for almost all $x\in (1,2)$. Then we prove that for all $\varepsilon>0$, there exists $x\in (1,\, 1+\varepsilon)$ and $r\in (2-\varepsilon,\, 2)$ such that $\mathbf S_x$ and $\mathbf S_r$ have a Sidon tail.