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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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Classification of tile digit sets as product-forms
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by Chun-Kit Lai, Ka-Sing Lau and Hui Rao PDF
Trans. Amer. Math. Soc. 369 (2017), 623-644 Request permission


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:,
  • Ka-Sing Lau
  • Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
  • MR Author ID: 190087
  • Email:
  • Hui Rao
  • Affiliation: Department of Mathematics, Central China Normal University, Wuhan, People’s Republic of China
  • Email:
  • 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).
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 623-644
  • MSC (2010): Primary 11B75, 52C22; Secondary 11A63, 28A80
  • DOI:
  • MathSciNet review: 3557788