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

Abstract

AbstractIn 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.