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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Spectral structure of digit sets of self-similar tiles on $ {\mathbb{R}}^1$


Authors: Chun-Kit Lai, Ka-Sing Lau and Hui Rao
Journal: Trans. Amer. Math. Soc. 365 (2013), 3831-3850
MSC (2010): Primary 11A63; Secondary 11B75, 28A80, 52C22
Published electronically: February 26, 2013
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Abstract: We study the structure of the digit sets $ {\mathcal D}$ for the integral self-similar tiles $ T(b,{\mathcal {D}})$ (we call such a $ {\mathcal D}$ a tile digit set with respect to $ b$). So far the only available classes of such tile digit sets are the complete residue sets and the product-forms. Our investigation here is based on the spectrum of the mask polynomial $ P_{\mathcal D}$, i.e., the zeros of $ P_{\mathcal D}$ on the unit circle. By using the Fourier criteria of self-similar tiles of Kenyon and Protasov, as well as the algebraic techniques of cyclotomic polynomials, we characterize the tile digit sets through some product of cyclotomic polynomials (kernel polynomials), which is a generalization of the product-form to higher order.


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Additional Information

Chun-Kit Lai
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
Address at time of publication: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
Email: cklai@math.cuhk.edu.hk, cklai@math.mcmaster.ca

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

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05787-X
PII: S 0002-9947(2013)05787-X
Keywords: Blocking, cyclotomic polynomials, kernel polynomials, prime, product-forms, self-similar tiles, spectra, tile digit sets, tree
Received by editor(s): March 31, 2011
Received by editor(s) in revised form: November 29, 2011
Published electronically: February 26, 2013
Additional Notes: The research was supported in part by the HKRGC Grant, the Direct Grant and the Focused Investment Scheme of CUHK
The third author was also supported by the National Natural Science Foundation of China (Grant Nos. 10501002 and 11171128).
Article copyright: © Copyright 2013 American Mathematical Society