A New Result on Packing Unit Squares into a Large Square

TL;DR

This paper improves the upper bound on the uncovered area when packing unit squares into a large square, and refines the minimum number of squares needed for coverage, advancing previous mathematical results.

Contribution

It presents a tighter bound of O(x^{5/8}) for the uncovered area and the covering number, improving upon earlier bounds by Chung and Graham and others.

Findings

Uncovered area bound improved to O(x^{5/8})

Minimum covering squares bound refined to x^2 + O(x^{5/8})

Advances previous packing and covering bounds in geometric combinatorics.

Abstract

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 $x^2+O(x^{5/8})$.