整数自仿Tiling的Tile数字集(英文)

Tile Digit Sets of Integral Self-affne Tilings

关于由一个扩张矩阵A∈Mn(Z)和数字集D={d1,d2,...,dm}■Zn生成的整数自仿Tiling,已经有很多研究结果。其中一个重要的问题是判定一个数字集在什么条件下能生成一个Tile。在一维情况下,已知结果有:标准数字集,乘积形式数字集,弱乘积形式数字集都是Tile数字集。在本文中,我们把弱乘积形式的概念推广到高维,并证明它们都是Tile数字集。

Integral self-affne tilings generated by an expanding integer matrix A ∈ Mn（Z） and D ={d1,d2,...,dm} lohtain in Zn have been studied by many works.An important problem is to decide when a digit set gives us a tile（and then we call it a tile digit set）.It is shown that the standard digit sets by Bandt,product form digit sets by Lagarias and Wang,and weak-product form digit sets in R1 by Lau and Rao are tile digit sets.In this paper,we generalize the notion of weak product form to higher dimensions and prove that they are tile digit sets.