  In this paper we investigate pigeonhole for the fractional parts of the
sequence $\sqrt{n}$. Namely, we partition the unit circle into N intervals and
investigate the limiting distribution of the first N points of the sequence
fractional parts of $\sqrt{n}$ among such a partition. In particular, we show
that the proportion of partition intervals containing exactly j points of our
sequence for each natural number j converges in the limit as N tends to
infinity.
  More generally, we investigate how the limiting distribution of the first
$sN$ points of the sequence varies with the parameter s>0. A natural way to
examine this is via point processes - random measures on the positive real
half-line which represent the arrival times of the points of our sequence to a
random interval from our partition. Indeed, we show that the sequence of point
processes we obtain converges in distribution to a point process on the
positive real half-line and give an explicit description of the limiting
process in terms of random affine unimodular lattices.
  Our work builds upon that of Elkies and McMullen and, explicitly, uses
ergodic theory on homogeneous space of affine unimodular lattices.
Specifically, we prove an equidistribution result about rational points along
non-linear horocycle sections - closed curves which restrict to give horocycles
in the modular surface and which equidistribute in the space when expanded.
This result is a generalisation the equidistribution of rational points on
expanding horocycles in the modular surface.
