# Weyl's equidistribution theorem in the case of rational numbers

Let θ be a non-zero real number and let N be a positive integer. Consider the fractional parts of nθ for 0 ≤ n < N. This creates a maximum of N possible intervals. Show that there as only three possible length of this interval. I have no idea how to do this problem. It looks so simple but I am really unable to proof it. So any help will be appreciated, thanks.

• How do the fractional parts of $n\theta$ create even one interval? Can you illustrate with an example? – David K Jun 30 at 18:29
• $\theta = 0.3$, $N = 5$, fractional parts are $0, 0.3, 0.6, 0.9, 0.2$. Three intervals that have different lengths are $(0,0.2), (0.2,0.3), (0.3,0.6)$. – peterwhy Jun 30 at 18:33
• The title is not good. – Dietrich Burde Jun 30 at 18:41
• @JeanMarie Ran a program for $\theta = \sqrt 2$ and $N=100$, the three distinct interval lengths (up to 8 decimal places) are 5.05063e-3,7.14267e-3,1.219331e-2. 7.14267e-3 happens in one interval only and is approximately the length of interval $(0, 99\sqrt 2\bmod 1) = (0, 99\sqrt 2 - 140)$. Also the shorter two lengths add up to the third. – peterwhy Jun 30 at 23:54
• This is the Steinhaus Three-Gap Theorem, which see. – Gerry Myerson Jul 6 at 3:43