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 $N$?

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    $\begingroup$ 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.


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