TITLE: Are gaps in $n^2\sqrt{2}$ poissonian?
QUESTION [7 upvotes]: I would like to know gaps about the sequence $n^2 \sqrt{2} \mod 1$.  Van der Corput's trick shows that $n^2 \sqrt{2}$ is equidistributed on the circle.  For large $N$, the fraction
$$ \frac{\# \{ 1 \leq i \leq N: a < n^2 \sqrt{2} \mod 1 < b \} }{N} \approx b-a.$$
Do the spacings between these values approach the Poisson distribution (as they would for uniformly random numbers)?  Is there a proof of this involving Homogenous spaces?  This is different from Elkies and McMullen's paper where they consider $\sqrt{n}\mod 1$ and relate it to the 5D space of Euclidean lattices.

REPLY [13 votes]: Rudnick and Sarnak (MR1628282) conjecture that any not-too-well-approximable $\alpha$ has this property in place of $\sqrt{2}$, in particular any algebraic number should have this property. They prove that almost every $\alpha$ (in the Lebesgue sense) have this property, but no specific $\alpha$ has been exhibited so far. Rudnick-Sarnak-Zaharescu (MR1839285) and Zaharescu (MR1957276) have examined what happens for too-well-approximable $\alpha$'s. The above mentioned papers and their AMS reviews should serve as good starting points.