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On one-dimensional self-similar tilings and $pq$-tiles


Authors: Ka-Sing Lau and Hui Rao
Journal: Trans. Amer. Math. Soc. 355 (2003), 1401-1414
MSC (2000): Primary 52C20, 52C22; Secondary 42B99
DOI: https://doi.org/10.1090/S0002-9947-02-03207-5
Published electronically: November 20, 2002
MathSciNet review: 1946397
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Abstract: Let $b \geq 2$ be an integer base, $\mathcal{D} = \{ 0, d_1, \cdots , d_{b-1}\} \subset \mathbb{Z}$ a digit set and $T = T(b, \mathcal{D})$the set of radix expansions. It is well known that if $T$ has nonvoid interior, then $T$ can tile $\mathbb{R}$ with some translation set $\mathcal{J}$ ($T$ is called a tile and $\mathcal{D}$ a tile digit set). There are two fundamental questions studied in the literature: (i) describe the structure of $\mathcal{J}$; (ii) for a given $b$, characterize $\mathcal{D}$ so that $T$ is a tile.

We show that for a given pair $(b,\mathcal{D})$, there is a unique self-replicating translation set $\mathcal{J} \subset \mathbb{Z}$, and it has period $b^m$ for some $m \in \mathbb{N}$. 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.


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Additional Information

Ka-Sing Lau
Affiliation: Department of Mathematics, the Chinese University of Hong Kong, Hong Kong
Email: kslau@math.cuhk.edu.hk

Hui Rao
Affiliation: Department of Mathematics and Nonlinear Science Center, Wuhan University, Wuhan, 430072, P.R. China; Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
Email: raohui@tsuda.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-02-03207-5
Received by editor(s): February 13, 2002
Received by editor(s) in revised form: September 11, 2002
Published electronically: November 20, 2002
Additional Notes: The authors are partially supported by an HKRGC grant and also a direct grant from CUHK. The second author is supported by CNSF 19901025.
Article copyright: © Copyright 2002 American Mathematical Society

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