Tiling the unit square with squares and rectangles (Q1062992)

Call a rectangle small if it will fit inside the unit square; call a rectangle binary if its dimensions are powers of 2. Theorem 1. If \({\mathcal S}\) is a set of small binary squares having total area greater than 1, some finite subset of \({\mathcal S}\) will tile the unit square. Theorem 2. If \({\mathcal R}\) is a set of small binary rectangles having total area greater than 5/3, some finite subset of \({\mathcal R}\) will tile the unit square. The above bounds are sharp. Various corollaries are drawn. In particular we show that if \({\mathcal R}\) is a collection of rectangles, then \({\mathcal R}\) can cover the plane iff \(\Sigma \{\min (a,1)\cdot \min (b,1)| R\in {\mathcal R}\}=\infty,\) where for a given rectangle, a is its width and b is its length.