Given any $\alpha,\beta\in (0, 1)$, $k\in Z^+, n > 1$ is this true ($\mathcal{F}_n$ denotes the $n$th Farey sequence, and $\mathcal{F_n}^{\prime} = \{q:q = a + b, a\in\mathbb{Z}, b\in\mathcal{F_n}\}$):
$\forall p, q\in\mathbb{Z}$, $$|(\alpha + p\beta, \alpha + (p + 1)\beta)\cap\mathcal{F}_n\prime|\sim|(\alpha + q\beta, \alpha + (q + 1)\beta)\cap\mathcal{F}_n\prime|$$
(Essentially, the cardinalities approach each other as $n$ grows larger)
I would preferably want a proof of this, or some hints as to how to prove this or generalize this.