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Weyl's criterion asserts that a real sequence $x_1, x_2, ...$ is equidistributed if and only if $$\lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^n e^{2 \pi i m x_n} = 0$$ for each integer $m \geq 1$.

One can for instance show using Weyl's criterion that a number $\vartheta$ is irrational if and only if the sequence $(n \vartheta)_{n \geq 1}$ is equidistributed.

I was wondering now whether this could be used to show irrationality of certain numbers.

Or asked differently: Equidistribution seems to me as a quite abstract thing to care about (that is: I have to convince myself that this is interesting). So why is equidistribution interesting from a number theoretic point of view?

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