# Density of the set of the fractional part of sufficiently large irrational numbers in the unit interval $[0,1]$

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 $[0,1]$?

• How about $x = 2$? – Nate Nov 10 '14 at 16:29
• I meant "$x \notin \mathbb{Q}$". Sorry! (Fixed in the question) – 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 $n\to\infty$.
• 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? – Steven Stadnicki Nov 10 '14 at 18:08