Piercing quasi-rectangles: On a problem of danzer and rogers

Abstract

It is an old problem of Danzer and Rogers to decide whether it is possible arrange O(1/ε) points in the unit square so that every rectangle of area ε 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 is Θ (1/ε log 1/ε).