Orbit of a point. $\newcommand{\Orb}{\operatorname{Orb}}$ Let $\Theta \in (0,1)$ be irrational. For all $x \in [0,1]$, define $$\Orb(x) = \\{ \\{x + n \Theta\\} : n \in \mathbb{Z} \\}$$ My question is: Is it true th...--prophetes.ai

Orbit of a point. $\newcommand{\Orb}{\operatorname{Orb}}$ Let $\Theta \in (0,1)$ be irrational. For all $x \in [0,1]$, define $$\Orb(x) = \\{ \\{x + n \Theta\\} : n \in \mathbb{Z} \\}$$ My question is: Is it true that $\Orb(x) = \Orb(y)$ OR $\Orb(x) \cap \Orb(y) = \varnothing$? I think this is true if we can show that $[0,1]$ can be partitioned into orbits of all of its points... Can someone help me to prove this in case this is true? thanks a lot.

**Hint:** Prove that $x\sim y$ iff $y\in Orb(x)$ defines an equivalence relation