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

Calculus Level 3

Let \(\alpha \in \mathbb{R}\) be an irrational number. Consider the set \[S := \{\{2^n \alpha\} \, | \, n \in \mathbb{N}\},\] where \(\{x\}\) denotes the fractional part of \(x\). Is \(S\) dense in \([0,1]\)?

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