Compactness Theorems for Geometric Packings

Moser asked whether the collection of rectangles of dimensions 1×12, 12×13, 13×14, …, whose total area equals 1, can be packed into the unit square without overlap, and whether the collection of squares of side lengths 12, 13, 14, … can be packed without overlap into a rectangle of area π2/6−1. Computational investigations have been made into packing these collections into squares of side length 1+ε and rectangles of area π2/6−1+ε, respectively, and one can consider the apparently weaker question of whether such packings are possible for every positive number ε. In this paper we establish a general theorem on sequences of geometrical packings that implies, in particular, that the “for every ε” versions of these two problems are actually equivalent to the original tiling problems.