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

Calculus Level 3

Let \(\alpha \in \mathbb{R}\) be an irrational number. Consider the set \[S := \big\{\{2^n \alpha\} \, | \, n \in \mathbb{N}\big\},\] where \(\{x\}\) denotes the fractional part of \(x\).

Is \(S\) dense in \([0,1]?\)