It seems intuitive to me that in the long run, the fractional part of any irrational number should cover the unit interval uniformly. More concretely, I'm wondering how one would prove the following, preferably in as simple terms as possible:

$\forall r,a,b$ s.t. $r \in \mathbb{R} \backslash \mathbb{Q}$ and $0 \le a \le b \le 1$,

$$\lim_{n=\infty} \frac{\sum_{i=1}^n \mathbb{1}_{\{ir\} \in [a,b]}}{n} = b-a$$

where $\{x\}$ is the fractional part of $x$.

It's easy to prove that $\{ir\}$ is dense in $[0,1]$ using Pigeonhole but proving uniformness is much harder.

Edit: I came across this statement when looking at Benford's Law. It turns out there is a fairly short proof of the fact that if your data-generating process is multiplying some initial number x by some chosen growth rate r for an arbitrary number of time periods, then you generate the Benford Law distribution assuming the statement above is true.


1 Answer 1


This is the Equidistribution theorem.


A proof is, for example, here:





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