Classification of tile digit sets as product-forms
Authors:
Chun-Kit Lai, Ka-Sing Lau and Hui Rao
Journal:
Trans. Amer. Math. Soc. 369 (2017), 623-644
MSC (2010):
Primary 11B75, 52C22; Secondary 11A63, 28A80
DOI:
https://doi.org/10.1090/tran/6703
Published electronically:
April 15, 2016
MathSciNet review:
3557788
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Abstract | References | Similar Articles | Additional Information
Abstract: Let $A$ be an expanding matrix on $\mathbb {R}^s$ with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set $\mathcal {D}\subset \mathbb {Z}^s$ so that the integral self-affine set $T(A,\mathcal D)$ is a translational tile on $\mathbb {R}^s$. In our previous paper, we classified such tile digit sets $\mathcal {D}\subset \mathbb {Z}$ by expressing the mask polynomial $P_{\mathcal {D}}$ as a product of cyclotomic polynomials. In this paper, we first show that a tile digit set in $\mathbb {Z}^s$ must be an integer tile (i.e., ${\mathcal D}\oplus {\mathcal L} = \mathbb {Z}^s$ for some discrete set ${\mathcal L}$). This allows us to combine the technique of Coven and Meyerowitz on integer tiling on $\mathbb {R}^1$ together with our previous results to characterize explicitly all tile digit sets $\mathcal {D}\subset \mathbb {Z}$ with $A = p^{\alpha }q$ ($p, q$ distinct primes) as modulo product-form of some order, an advance of the previously known results for $A = p^\alpha$ and $pq$.
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Additional Information
Chun-Kit Lai
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada
Address at time of publication:
Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, California 94132
MR Author ID:
950029
Email:
cklai@math.mcmaster.ca, cklai@sfsu.edu
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, Central China Normal University, Wuhan, People’s Republic of China
Email:
hrao@mail.ccnu.edu.cn
Keywords:
Blocking,
cyclotomic polynomials,
integer tiles,
kernel polynomials,
prime,
product-forms,
self-affine tiles,
spectra,
tile digit sets,
tree
Received by editor(s):
May 1, 2013
Received by editor(s) in revised form:
January 14, 2015
Published electronically:
April 15, 2016
Additional Notes:
This research was supported in part by the HKRGC grant and the NNSF of China (Nos. 11171100, 11371382).
Article copyright:
© Copyright 2016
American Mathematical Society