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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.

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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.

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