## 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