Rational points on nonlinear horocycles and pigeonhole statistics for the fractional parts of

Abstract

In this paper, we investigate pigeonhole statistics for the fractional parts of the sequence $\sqrt {n}$ . Namely, we partition the unit circle $ \mathbb {T} = \mathbb {R}/\mathbb {Z}$ into N intervals and show that the proportion of intervals containing exactly j points of the sequence $(\sqrt {n} + \mathbb {Z})_{n=1}^N$ converges in the limit as $N \to \infty $ . More generally, we investigate how the limiting distribution of the first $sN$ points of the sequence varies with the parameter $s \geq 0$ . A natural way to examine this is via point processes—random measures on $[0,\infty )$ which represent the arrival times of the points of our sequence to a random interval from our partition. We show that the sequence of point processes we obtain converges in distribution and give an explicit description of the limiting process in terms of random affine unimodular lattices. Our work uses ergodic theory in the space of affine unimodular lattices, building upon work of Elkies and McMullen [Gaps in $\sqrt {n}$ mod 1 and ergodic theory. Duke Math. J.123 (2004), 95–139]. We prove a generalisation of equidistribution of rational points on expanding horocycles in the modular surface, working instead on nonlinear horocycle sections.

Smallest denominators

“…There has been significant interest recently in studying the distribution of rational points on horospheres. We refer the interested reader to [4,[7][8][9]22] and references therein.…”

“…There has been significant interest recently in studying the distribution of rational points on horospheres. We refer the interested reader to [4, 7–9, 22] and references therein. Proposition For $$f:{\mathbb {T}}^n\times \Gamma \backslash G\rightarrow {\mathbb {R}}$$ bounded continuous, $$c>0$$, we have $$$$\begin{equation} \lim _{\substack{N,Q\rightarrow \infty cQ^{n+1}\leqslant N^n}} \frac{1}{N^n} \sum _{{\text{\boldmath $j$}}\in {\mathbb {Z}}^n/N{\mathbb {Z}}^n} f{\left(N^{-1}{\text{\boldmath $j$}},h({\text{\boldmath $x$}}_0+N^{-1}{\text{\boldmath $j$}})a(Q)\right)} = \int _{{\mathbb {T}}^n\times \Gamma \backslash G} f({\text{\boldmath $x$}},g) \, d{\text{\boldmath $x$}}\, d\mu (g).$$…”

“…The role of the void statistics for Farey fractions is now replaced by the so-called pigeonhole statistics, which is of independent interest and the content of Section 4. We will use a similar strategy of proof as recently employed by Pattison [22] for the pigeonhole statistics of √ 𝑛 mod 1. In Section 5, we discuss moments of the distance function for the multidimensional Farey sequence, thus extending results of Kargaev and Zhigljavsky [12].…”