Weyl's theorem on equal distribution

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The set of Weyl (by Hermann Weyl ) is the basis for arithmetic random number generators . It says:

Be an irrational number . Then it has the consequence ${\displaystyle y_{0}\in \ ]0,1[}$

${\displaystyle (u_{i})_{i\geq 1}\subseteq \ ]0,1[}$,

defined by term by

${\displaystyle u_{i}=iy_{0}-\lfloor iy_{0}\rfloor =iy_{0}\ {\bmod {\ }}1}$

the asymptotic uniform distribution property. So for everyone with : ${\displaystyle a,b\in \mathbb {R} }$ ${\displaystyle 0<a<b<1}$

${\displaystyle {\frac {\left|\{i|1\leq i\leq n;a\leq u_{i}\leq b\}\right|}{n}}\quad {\xrightarrow[{n\rightarrow \infty }]{\quad }}\quad b-a}$.

In other words: the probability that an arbitrarily chosen sequence term lies in is . ${\displaystyle [a,b]}$ ${\displaystyle b-a}$