Spectral structure of digit sets of self-similar tiles on

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

DOI:
https://doi.org/10.1090/S0002-9947-2013-05787-X

Published electronically:
February 26, 2013

MathSciNet review:
3042605

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the structure of the digit sets for the integral self-similar tiles (we call such a a *tile digit set* with respect to ). 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 , i.e., the zeros of 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:
https://doi.org/10.1090/S0002-9947-2013-05787-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