Perfect packings of squares using the stack-pack strategy
A packing of an infinite collection of square tiles of bounded total area into a finite number of rectangles is called perfect when the total area of the rectangles is the same as that of the squares. The author considers the special problem of perfectly packing the collection of squares of edge lengths \(\{n^{-t}:n\geq1\}\), for fixed \(t\in(0,1)\), into a set of \(m\) square boxes, where \(m=m(t) < \infty\) is not otherwise bounded a priori. He proves that if \(1/2 < t < 2/3\), then this is indeed possible: There exists \(m(t)<\infty\) such that after packing each of the first \(m(t)-1\) squares into a new separate box, the remaining squares may be perfectly packed into one additional square. The author does not give an explicit bound on \(m(t)\). There are some misprints in the paper; namely, Theorem 3.2 should start with ``If \(t>1/2\)\dots'', and Theorem 4.1 with ``If \textbf{B} is a finite collection \dots''.
scientific article; zbMATH DE number 1969107