# 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 $$[0,1]?$$

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