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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On one-dimensional self-similar tilings and $pq$-tiles
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by Ka-Sing Lau and Hui Rao PDF
Trans. Amer. Math. Soc. 355 (2003), 1401-1414 Request permission

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
  • MR Author ID: 190087
  • 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
  • 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.
  • © Copyright 2002 American Mathematical Society
  • 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
  • MathSciNet review: 1946397