## Disklikeness of planar self-affine tiles

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- by King-Shun Leung and Ka-Sing Lau PDF
- Trans. Amer. Math. Soc.
**359**(2007), 3337-3355 Request permission

## Abstract:

We consider the disklikeness of the planar self-affine tile $T$ generated by an integral expanding matrix $A$ and a consecutive collinear digit set ${\mathcal {D}}= \{0, v, 2v, \cdots , (|q|-1)v \}\subset {\Bbb {Z}}^2$. Let $f(x)=x^{2}+ p x+ q$ be the characteristic polynomial of $A$. We show that the tile $T$ is disklike if and only if $2|p|\leq |q+2|$. Moreover, $T$ is a hexagonal tile for all the cases except when $p=0$, in which case $T$ is a square tile. The proof depends on certain special devices to count the numbers of*nodal points*and

*neighbors*of $T$ and a criterion of Bandt and Wang (2001) on disklikeness.

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## Additional Information

**King-Shun Leung**- Affiliation: Department of Mathematics, Science, Social Sciences and Technology, The Hong Kong Institute of Education, Tai Po, Hong Kong
- Email: ksleung@ied.edu.hk
**Ka-Sing Lau**- Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
- MR Author ID: 190087
- Email: kslau@math.cuhk.edu.hk
- Received by editor(s): October 14, 2004
- Received by editor(s) in revised form: June 23, 2005
- Published electronically: February 13, 2007
- Additional Notes: This research was partially supported by an HK RGC grant
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**359**(2007), 3337-3355 - MSC (2000): Primary 52C20, 52C22; Secondary 28A80
- DOI: https://doi.org/10.1090/S0002-9947-07-04106-2
- MathSciNet review: 2299458