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.

  • $\begingroup$ Why is the denominator $\log K$ and not just $K$? $\endgroup$ – Erick Wong Sep 7 '17 at 23:16
  • 2
    $\begingroup$ Because $\sum_{n=1}^K (b/n - a/n) \approx (b - a) \log K$. $\endgroup$ – Mateusz Kwaśnicki Sep 7 '17 at 23:26

The solution should be $b-a$. I don't know how difficult this is to prove from scratch, but I think it follows for example from work of W.M. Schmidt, see: Schmidt, Wolfgang M., A metrical theorem in geometry of numbers. Trans. Amer. Math. Soc. 95, 1960 516–529. This is also contained as Theorem 4.1 in Harman's book on metric number theory. (In this setting, the result is stated for counting $\|n \theta\| \leq \psi(n)$ for some function $\psi$, where $\| \cdot \|$ is the distance to the nearest integer, but you should be able to translate this to your question, I guess.)


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