On one-dimensional self-similar tilings and $pq$-tiles
On one-dimensional self-similar tilings and $pq$-tiles
复制标题
DOI：
10.1090/s0002-9947-02-03207-5
复制
发表时间：
2002-11
影响因子：
1.3
通讯作者：
K. Lau;H. Rao
中科院分区：
文献类型：
--
作者：
K. Lau;H. Rao
设B > 2为整数基，D = {0，d 1，.，d B-1 } n = Z是一个数字集，T = T（B，D）是基数扩展集。众所周知，如果T具有非空内部，则T可以将R与某个平移集J拼接（T称为拼接，D称为拼接数集）。有两个基本的问题，在文献中研究：（i）描述J的结构;（ii）对于一个给定的B，表征D，使T是一个瓦片。我们证明了对于给定的对（B，D），存在唯一的自复制平移集JC Z，并且它的周期为B m，其中m ∈ N.这完成了凯尼恩早期的一些工作。我们的主要结果（ii）是刻画了当p，q是不同素数时B = pq的瓦片数集。唯一的其他已知的特征是B = pl，由于Lagarias和王。pq情形的证明依赖于Kenyon和De Bruijn关于分圆多项式的技巧，以及Odlyzko乘积形式数集的扩展。
Let b > 2 be an integer base, D = {0, d 1 , ..., d b-1 } ⊂ Z a digit set and T = T(b, D) the set of radix expansions. It is well known that if T has nonvoid interior, then T can tile R with some translation set J (T is called a tile and D a tile digit set). There are two fundamental questions studied in the literature: (i) describe the structure of J; (ii) for a given b, characterize D so that T is a tile. We show that for a given pair (b, D), there is a unique self-replicating translation set J C Z, and it has period b m for some m ∈ N. This completes some earlier work of Kenyon. Our main result for (ii) is to characterize the tile digit sets for b = pq when p, q are distinct primes. The only other known characterization is for b = p l , due to Lagarias and Wang. The proof for the pq case depends on the techniques of Kenyon and De Bruijn on the cyclotomic polynomials, and also on an extension of the product-form digit set of Odlyzko.