Consider the covering of the plane by the grid squares, that is, by the unit squares whose vertices are the grid points.
Prove that there is an absolute constant $C > 0$ so that for every $0.1 > ε > 0$ there is a set $F$ of points in the plane containing exactly one point in each grid square so that every planar (not necessarily aligned) rectangle of dimensions $\epsilon \times C \frac 1 \epsilon \log(\frac 1 \epsilon) $, contains at least one point of $F$.