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For $A=\{\lfloor n \alpha \rfloor: n\in\Bbb Z \}$, where $\alpha$ irrational, $\alpha \gt 2$, we aim to show the following:

  • There exists $m$ elements contained in $A$ that form an arithmetic progression, for any $m \gt 2$, $m \in\Bbb N$.

  • There exists no infinite arithmetic progression in $A$.

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