Packing anchored rectangles

TL;DR

The paper constructs a set of interior-disjoint axis-aligned empty rectangles anchored at points in a unit square, covering a positive constant area, advancing towards a conjecture that such packings can cover at least half the square.

Contribution

It provides a construction demonstrating that a set of anchored empty rectangles can cover a positive constant area, moving closer to proving the conjecture of covering at least half the square.

Findings

Constructed a packing covering about 0.09 of the area

Proved the existence of such packings for any point set

Progressed towards the conjecture of covering at least 1/2 of the square

Abstract

Let $S$ be a set of $n$ points in the unit square $[0,1]^2$, one of which is the origin. We construct $n$ pairwise interior-disjoint axis-aligned empty rectangles such that the lower left corner of each rectangle is a point in $S$, and the rectangles jointly cover at least a positive constant area (about 0.09). This is a first step towards the solution of a longstanding conjecture that the rectangles in such a packing can jointly cover an area of at least 1/2.