Let $\langle x \rangle: \mathbb{R} \to (-1/2,1/2]$ be the periodic function with period $1$ which is $x$ for $x \in (-1/2,1/2]$. Is there some function $D(a,b)$ of real numbers $a<b$ such that, for almost all $\theta \in \mathbb{R}$ and all $a<b$, we have $$D(a,b) = \lim_{K \to \infty} \frac{1}{\log K} \#{\Big \{} n : \langle n \theta \rangle \in (a/n,b/n),\ 1 \leq n \leq K {\Big \}}.$$ If so, what is $D$?

I know about the Gauss-Kuzmin distribution, but it wasn't clear how to me extract this from that G-K distribution .

**EDIT** After thinking a bit, the most likely guess for $D(a,b)$ is simply $b-a$. But it isn't so clear how to prove it.