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Báo cáo toán học: "Filling a box with translates of two bricks"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Filling a box with translates of two bricks. | Filling a box with translates of two bricks Mihail N. Kolountzakis School of Mathematics Georgia Institute of Technology 686 Cherry Street NW Atlanta GA 30332 USA and Department of Mathematics University of Crete Knossos Ave. 714 09 Iraklio Greece. E-mail kolount@member.ams.org Received Sep 20 2004 Accepted Dec 1 2004 Published Dec 7 2004 Mathematics Subject Classifications 05B45 52C22 Abstract We give a new proof of the following interesting fact recently proved by Bower and Michael if a d-dimensional rectangular box can be tiled using translates of two types of rectangular bricks then it can also be tiled in the following way. We can cut the box across one of its sides into two boxes one of which can be tiled with the first brick only and the other one with the second brick. Our proof relies on the Fourier Transform. We also show that no such result is true for translates of more than two types of bricks. Suppose we have at our disposal two types of d-dimensional rectangles bricks type A with dimensions a1 . ad and type B with dimensions b1 . bd . We want to use translates of such bricks to fill completely and with no overlaps except at the boundaries of the bricks a given d-dimensional rectangular box. We then say that these two bricks tile our box by translations. All rectangles that appear in this note are axis-aligned. Bower and Michael 1 recently showed the following nice result. A hyperplane cut is a separation of an axis-aligned box in d dimensions using a hyperplane of the type Xj a Supported in part by European Commission IHP Network HARP Harmonic Analysis and Related Problems Contract Number HPRN-CT-2001-00273 - HARP. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 N16 1 for some j 1 . d and some a 2 R. A hyperplane cut separates such a box into two rectangular boxes. Theorem 1 Bower and Michael 1 If two bricks of types A and B tile a box Q in dimension d 1 by translations then we can split Q into two other boxes Qa and Qb using a hyperplane cut such .