Rational points on non-linear horocycles and pigeonhole statistics for the fractional parts of $\sqrt{n}$

In this paper we investigate $\mathit{\text{pigeonhole}}\mathit{\text{statistics}}$ for the fractional parts of the sequence $\sqrt{n}$. Namely, we partition the unit circle $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$ 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 $N\to \mathrm{\infty }$. More generally, we investigate how the limiting distribution of the first $sN$ points of the sequence varies with the parameter $s\ge 0$. A natural way to examine this is via point processes - random measures on $[0,\mathrm{\infty })$ 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. We prove a generalisation of equidistribution of rational points on expanding horocycles in the modular surface, working instead on non-linear horocycle sections.