Piercing quasi-rectangles-On a problem of Danzer and Rogers

It is an old problem of Danzer and Rogers to decide whether it is possible to arrange O(1ε) points in the unit square so that every rectangle of area ε>0 within the unit square contains at least one of them. We show that the answer to this question is in the negative if we slightly relax the notion of rectangles, as follows.Let δ be a fixed small positive number. A quasi-rectangle is a region swept out by a continuously moving segment s, with no rotation, so that throughout the motion the angle between the trajectory of the center of s and its normal vector remains at most δ. We show that the smallest number of points needed to pierce all quasi-rectangles of area ε>0 within the unit square is Θ(1εlog1ε).