Let $a$ an irrational number. Can we say that there is $c>0$ such that for all integer $k,n$ and $1>u>0$, $$ \frac {1}{n}\sum_{i=k}^{k+n} 1_{( (ia) \in [0,u])}< cu ? $$ where (x) is the fractional part of a real number $x$.

It might be related to the fact that the sequence $\{an\}$ is "well-distributed": for all interval $I$ in $[0,1]$, $$

\frac 1n \sum_{i=k}^{k+n} 1_{(ia)\in I}\to |I| $$ "uniformly over k", as $n\to \infty$. I think the "uniformly over k" means that for $I$ fixed, the distance to the limit can be bounded only by means of n. The problem for me is that it is not "uniformly over $ I$" (or at least over $I=[0,u]$).

EDIT: I found a partial answer (with a power of u larger than 1 on the RHS) but I'm still open to a solution! I also posted the following related question: Very badly approximable numbers