Is $\{2^n \alpha\}$ A Dense Set?

Calculus Level 3

Is the following statement true for all irrational numbers $\alpha$ :

Consider the set $S := \big\{\{2^n \alpha\} \, | \, n \in \mathbb{N}\big\},$ where $\{x\}$ denotes the fractional part of $x$ . Then, $S$ is dense in $[0,1]$ .

Is {2nα}\{2^n \alpha\}{2nα} A Dense Set?

Is $\{2^n \alpha\}$ A Dense Set?