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I have two irrational rotations $R_1x=x+\alpha$, $R_2x=x+\beta$ which are isomorphic. I need to prove that there exists $a,b \in \mathbb{N}$ such that $a\alpha+b\beta \in \mathbb{Z}$. How can I do that?

Mayers
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  • Presumably you're thinking of translations as "rotations" on the circle $\mathbf{R}/\mathbf{Z}$...? Could you please clarify this point, and say what "isomorphic" means for rotations? ("Topologically conjugate"? "Generate the same cyclic subgroup of transformations"? Something else...?) – Andrew D. Hwang Nov 24 '16 at 13:25
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    Yes, I consider rotation on the circle $\mathbb{R}/\mathbb{Z}$ which I identify with interval $[0,1)$. My isomorphism is defined as follows: two systems $([0,1),\mathcal{B}{[0,1)}, \lambda, R_1)$ and $([0,1),\mathcal{B}{[0,1)}, \lambda, R_2)$ are isomorphic if there are sets $X \subset [0,1)$ and $Y\subset [0,1)$ of measure one and a measure-preserving transformation $\phi: X\to Y$ such that $\phi \circ R_1 = R_2\circ \phi$ on $X$ and also $\lambda(\phi^{-1}Z)=\lambda(Z)$ for all measurable $Z\subset Y$. – Mayers Nov 24 '16 at 13:40

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