  In this paper we investigate \textit{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. We prove a generalisation of
equidistribution of rational points on expanding horocycles in the modular
surface, working instead on non-linear horocycle sections.
