The question is about the existence of a number $x$ for which we know the existence of $c>0$ such that for all $u>0,n\in\mathbb{N}^*$ that $$ \frac {1}{nu}\sum_{j=1}^{n}1_{d(jx,\mathbb Z)<u}<c $$ This holds for each $u>0$ for $x$ irrational (i.e. with $c$ depending on $u$, see references on "well distributed numbers"), but not uniformy in $u$.

Ideally it would be great to show that the series above is uniformly bounded from below by some $a>0$.

EDIT: the original statement was probably false, as noticed Anthony Quas, so I weakened it.