In their 2009 note: \emph{ Packing equal squares into a large square}, Chung and Graham proved that the uncovered area of a large square of side length $x$ is $O\left(x^{ (3+\sqrt{ 2})/7}\log x\right)$ after maximum number of non-overlapping unit squares are packed into it, which improved the earlier results of Erd\H{ o}s-Graham, Roth-Vaughan, and Karabash-Soifer. Here we further improve the result to $O(x^{ 5/8})$ that also helps to improve the bound for the dual problem: finding the minimum number of unit squares needed for covering the large square, from $x^2+O\left(x^{ (3+\sqrt{ 2})/7}\log x\right)$ to $x^2+O(x^{ 5/8})$.
Categories: math.CO
