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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.