Can we deduce that $⌊r^{n}α⌋≃r^{n}α$ when $r→∞$?

Question

Let $α∈(0,1)$ be an irrational number and let $n≥1$ be a fixed positive integer. For any $r>4$ we define the positive integer $$k=⌊r^{n}α⌋$$ where $⌊.⌋$ denotes the floor function.

My question is: Can we deduce that $$⌊r^{n}α⌋≃r^{n}α$$ when $r→∞$? (They have the same order of magnitude)

Answer 1

Yes, because $$1-\frac{1}{r^n\alpha}<\frac{\lfloor r^n \alpha\rfloor}{r^n\alpha}\leq 1$$