I've started reading the article "ɛ-nets and simplex range queries" and I found a weaker claim that i've been told can be proved more easily:
Prove that there is a constant $C > 0$ so that for every sufficiently small $\epsilon > 0$, one can choose exactly one point inside each grid square $[n, n + 1) × [m, m + 1) ⊂ \mathbb{R}^2 , m, n \in \mathbb{Z}$, so that every rectangle of dimensions $\epsilon \times (\frac{C}{\epsilon}) log(\frac{1}{\epsilon})$ in the plane (not necessarily axis-aligned) contains at least one chosen point.
It was hinted that the problem can be solved using the Local Lemma however I can't see how. Using compactness arguments I understand that it suffice to prove this for a bounded grid, but I don't understand how can the events could be defined.
ANY help would be appreciated