Is {2nα}\{2^n \alpha\} A Dense Set?

Calculus Level 3

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

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

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