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Transactions of the American Mathematical Society

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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
Published electronically: April 15, 2016
MathSciNet review: 3557788
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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

Ka-Sing Lau
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong

Hui Rao
Affiliation: Department of Mathematics, Central China Normal University, Wuhan, People’s Republic of China

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

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