Weyl criterion

A fundamental criterion used to solve the problem of the uniform distribution modulo one of an infinite sequence $(x_n)$ of arbitrary real numbers $x_n$ modulo 1, i.e. to establish that the limit as $N \rightarrow \infty$ of

$$ \sum_{n \le N : \alpha \le \{x_n\} \le \beta} \frac{1}{N} $$

exists and is equal to $\beta - \alpha$, where $ 0 \le \alpha \le \beta \le 1 $ and $\{x_n\}$ is the fractional part of $x_n$ (cf. Fractional part of a number). Weyl's criterion states that the sequence $(x_n)$ is uniformly distributed modulo 1 if and only if

$$ \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N \exp(2\pi i m x_n) = 0 $$

for all integers $m \ne 0$. Demonstrated in 1916 by H. Weyl. See Weyl method.

References

[1]	J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge University Press (1957)
[2]	H. Weyl, "Ueber die Gleichverteilung von Zahlen mod. Eins,". Math. Ann. 77, no.3 (1916) 313–352. DOI 10.1007/BF01475864