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In this paper we investigate \textit{pigeonhole statistics} for thefractional parts of the sequence $\sqrt{n}$. Namely, we partition the unitcircle $ \mathbb{T} = \mathbb{R}/\mathbb{Z}$ into $N$ intervals and show thatthe 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 wayto examine this is via point processes - random measures on $[0,\infty)$ whichrepresent the arrival times of the points of our sequence to a random intervalfrom our partition. We show that the sequence of point processes we obtainconverges in distribution and give an explicit description of the limitingprocess 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 ofequidistribution of rational points on expanding horocycles in the modularsurface, working instead on non-linear horocycle sections.

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