Let $\alpha$ be an irrational number, consider the basic dynamical system $T^{n}(0) = \{n \alpha\}$ where $\{.\}$ denotes the fractional part. Let $a < b$ be two numbers in $[0, 1]$. Then by Birkhoff's ergodic theorem and a little unique ergodicity argument we can prove that $$\operatorname{lim}_{N\rightarrow \infty} \frac{1}{N} \sum_{i = 0}^{N-1} \chi(T^{i}(0)) = \frac{1}{N} \sum_{i = 0} ^{N-1} \chi(\{i \alpha\}) = \int_{[0,1]} \chi=b-a $$ where $\chi$ is the characteristic function of the interval $[a, b]$. My question is that what can be said about the rate of this convergence. More specifically how large can be the discrepancy term $|\sum_{i=0}^{N-1} \chi(\{i \alpha\}) - N(b - a)|$ as a function of $N$?

  • 1
    $\begingroup$ This is very tricky. Check the book Irregularities of Distribution by Beck and Chen and also W.M Schmidt's articles with the same name. $\endgroup$ Sep 25, 2015 at 13:43

1 Answer 1


For a detailed account of this well-studied problem see, for example, Chapter 2 of Uniform Distribution of Sequences by L. Kuipers and H. Niederreiter.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.