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A colleague of mine asked me if the behavior of the sequence $\{\sin (n^2) \}_n$ is known. In particular, does it converge? If not, what are its liminf and limsup?

I had to admit that I cannot answer these questions, which are probably non-trivial.

Where can I find some hints or a discussion about these oscillating sequences?

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$\limsup a_n\neq \liminf a_n$, therfore $\lim{a_n}$ is not exist(Why we need Weyl's inequality ?) –  Salech Alhasov Oct 17 '12 at 8:57

2 Answers 2

up vote 2 down vote accepted

In virtue of the Weyl's inequality for quadratic polynomials and the Weyl's equidistribution Theorem, $e^{in^2}$ is equidistributed in the unit circle, so the sequence does not converge and limsup/liminf are $\pm 1$.

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Very interesting, I did not know enough about equidistributed sequences. I've just found this nice link: individual.utoronto.ca/hannigandaley/equidistribution.pdf –  Siminore Oct 17 '12 at 9:53

This has to do with the sequence {n^2/(2\pi} being equidistributed, see Weyl's criterion. In particular the liminf is -1 and the limsup is 1 as you would expect

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