## Classification of tile digit sets as product-forms

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**369**(2017), 623-644 Request permission

## 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 $pq$.## References

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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
- 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: https://doi.org/10.1090/tran/6703
- MathSciNet review: 3557788