# Covering a non-closed interval by disjoint closed intervals

this note will talk about the Ostrowski representation and approximation by continue fraction.

As well-known,by the Weyl criterion,$latex \{n\alpha\}$ is uniformly distribution in $latex [0,1]$ iff $latex \alpha\in R-Q$.

i.e. we have:$latex \forall 0\leq a\leq b\leq 1$,we have:

$latex \lim_{N\to \infty}|\{1\leq n\leq N|\{n\alpha\}\in [a,b]\}|=(b-a)N+o(N)$.

but this will not give the effective version.i.e. we do not the the more information about the decay of $latex o(N)$.

we will give a approach of effective version of $latex \alpha$ with smooth condition by give another proof of the uniformly distribution (in fact to to decomposition the interval $latex [a,b]$ in to a finite sums of special intervals).and get the result:

$latex D_N=\int_{M}D_N(\theta)d\mu=\int_Msup_{0<a<b<1}|\sum_{n=1}^{N}\chi_{(a,b)}(\{\theta n \})-N(b-a)|d\mu\sim O(log N)$

if the term in the continuous fraction of $latex \alpha$ have a up bound.this is so called $latex \alpha$ is smooth.