Elsevier

Discrete Mathematics

Volume 309, Issue 9, 6 May 2009, Pages 2913-2921
Discrete Mathematics

Rectangles as sums of squares

https://doi.org/10.1016/j.disc.2008.07.028Get rights and content
Under an Elsevier user license
open archive

Abstract

In this paper we examine generalisations of the following problem posed by Laczkovich: Given an n×m rectangle with n and m integers, it can be written as a disjoint union of squares; what is the smallest number of squares that can be used? He also asked the corresponding higher dimensional analogue. For the two dimensional case Kenyon proved a tight logarithmic bound but left open the higher dimensional case. Using completely different methods we prove good upper and lower bounds for this case as well as some other variants.

Keywords

Rectangle
Square
Discrepancy

Cited by (0)