My question is about a point process that I feel it would be natural to study, but that I have never heard of… This point process would represent, morally, the local behaviour of the set of fractions with bounded (but large) denominator, when seen from a random point. I would like to know if any of you has already heard about such a process…?
Let me give a formal definition of the point process that I would like to know more about. For $n$ a (large) integer, denote by $\mathbf{Q}_n$ the set of fractions with denominator $\leq n$: $$\mathbf{Q}_n := \bigl\{x \in \mathbf{R} \,\big|\, (\exists q \in \{1, \ldots, n\}) \ q x \in \mathbf{Z}\bigr\}\text.$$ So, morally this is the set of “simple enough” fractions having a bounded denominator.
You can look at $\mathbf{Q}_n$ from a (uniform) random point of the real line (note that $\mathbf{Q}_n$ is invariant by $1$-translations, so the notion of “uniform random point on the line” makes sense in this context): namely, for $X_0$ a real random variable uniform on $[0, 1)$, consider the point process defined by $$\mathfrak{N}_n := n^2 \times (\mathbf{Q}_n - X_0)$$ (here I see $\mathfrak{N}_n$ as a random variable whose values are discrete subsets of $\mathbf{R}$), whose law is completely well-defined. The factor $n^2$ in the definition of the process $\mathfrak{N}_n$ means that we are “zooming” on our set of simple fractions in such a way that the density of points remains of order $O(1)$ regardless of the value of $n$ (as it is a classical result that the density of points in $\mathbf{Q}_n$ is equivalent to $3 n^2 / \pi^2$ when $n \to \infty$).
Now, I have strong reasons to expect that when $n \to \infty$, the law of the process $\mathfrak{N}_n$ converges (for some appropriate topology on discrets subsets of $\mathbf{R}$) to the law of a limiting point process $\mathfrak{N}_\infty$. This process would somehow represent the local behaviour of the set of not-too-complicated fractions, when seen from a random point of the line. Note that one funny property of this point process in that two distinct points of it must always be apart from a distance $\geq 1$! (because, for $p / q$ and $p' / q'$ two dinstinct fractions of $\mathbf{Q}_n$, $\left|p / q - p' / q'\right| \geq 1 / q q' \geq n^{-2}$).
My question is: has anyone studied this process? If yes, how is this process called, and could you give me references on it?…
