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The van der Corput Lemma states

Van der Corput Lemma: Let $(x_n)$ be a bounded sequence in a Hilbert space $H$. Define a sequence $(s_n)$ by $s_h = \limsup_{N \to \infty} \left | \frac1N \sum_{n = 1}^N \langle x_{n + h}, x_h \rangle \right |.$ If there now holds that $\lim_{H \to \infty} \frac1H \sum_{h = 0}^{H - 1} s_h = 0,$ then we have that $\lim_{N \to \infty} \left \| \frac1N \sum_{n = 1}^N x_n \right \| = 0.$

We should be able to prove using this lemma that ($\{x\}$ denotes the fractional part of $x$) $\{n^2 \alpha \}$ is equidistributed where $\alpha$ is irrational.

Does someone have a hint how to do this? If I solve it I will modify my question to give the full solution. I assume that I am missing something simple.

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    Don't forget to use Weyl's criterion.2011-05-29

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Hint: Take $\displaystyle x_n (t) = e^{2\pi i n^2 \alpha t}$ in $L^2(S^1)$.

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    @Jonas: Well, I feel that accept was rather undeserved. Could you tell me what the Hilbert space is, since I haven't yet figured it out (my notes on that contain a really stupid mistake that make the whole argument nonsensical).2011-06-08