Is it true that $\forall x \notin \mathbb{Q}: x>1$, the set $A=\{ \operatorname{frac}(x^n): n \in \mathbb{N} \}$ is dense in $[0,1]$?

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    $\begingroup$ How about $x = 2$? $\endgroup$ – Nate Nov 10 '14 at 16:29
  • $\begingroup$ I meant "$x \notin \mathbb{Q}$". Sorry! (Fixed in the question) $\endgroup$ – user173897 Nov 10 '14 at 16:35

One of the questions in the first competition I did was to prove that $\{(2+\sqrt{3})^n\}\to1$ as $n\to\infty$.

  • $\begingroup$ Hint for anyone who wants to try this at home: what can you say about $(2+\sqrt{3})^n+(2-\sqrt{3})^n$? Can you see why this implies the result? $\endgroup$ – Steven Stadnicki Nov 10 '14 at 18:08

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