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

Published electronically:
February 26, 2013

MathSciNet review:
3042605

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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.

**[AT]**Shigeki Akiyama and Jörg M. Thuswaldner,*A survey on topological properties of tiles related to number systems*, Geom. Dedicata**109**(2004), 89–105. MR**2113188**, 10.1007/s10711-004-1774-7**[B]**Christoph Bandt,*Self-similar sets. V. Integer matrices and fractal tilings of 𝑅ⁿ*, Proc. Amer. Math. Soc.**112**(1991), no. 2, 549–562. MR**1036982**, 10.1090/S0002-9939-1991-1036982-1**[CM]**Ethan M. Coven and Aaron Meyerowitz,*Tiling the integers with translates of one finite set*, J. Algebra**212**(1999), no. 1, 161–174. MR**1670646**, 10.1006/jabr.1998.7628**[deB]**N. G. de Bruijn,*On the factorization of cyclic groups*, Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes Math.**15**(1953), 370–377. MR**0059271****[DFW]**Xin-Rong Dai, De-Jun Feng, and Yang Wang,*Refinable functions with non-integer dilations*, J. Funct. Anal.**250**(2007), no. 1, 1–20. MR**2345903**, 10.1016/j.jfa.2007.02.005**[GH]**Karlheinz Gröchenig and Andrew Haas,*Self-similar lattice tilings*, J. Fourier Anal. Appl.**1**(1994), no. 2, 131–170. MR**1348740**, 10.1007/s00041-001-4007-6**[H]**John E. Hutchinson,*Fractals and self-similarity*, Indiana Univ. Math. J.**30**(1981), no. 5, 713–747. MR**625600**, 10.1512/iumj.1981.30.30055**[HL]**Xing-Gang He and Ka-Sing Lau,*Characterization of tile digit sets with prime determinants*, Appl. Comput. Harmon. Anal.**16**(2004), no. 3, 159–173. MR**2054276**, 10.1016/j.acha.2004.03.001**[HLR]**Xing-Gang He, Ka-Sing Lau, and Hui Rao,*Self-affine sets and graph-directed systems*, Constr. Approx.**19**(2003), no. 3, 373–397. MR**1979057**, 10.1007/s00365-002-0515-0**[K]**Richard Kenyon,*Self-replicating tilings*, Symbolic dynamics and its applications (New Haven, CT, 1991) Contemp. Math., vol. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 239–263. MR**1185093**, 10.1090/conm/135/1185093**[KL1]**Ibrahim Kirat and Ka-Sing Lau,*On the connectedness of self-affine tiles*, J. London Math. Soc. (2)**62**(2000), no. 1, 291–304. MR**1772188**, 10.1112/S002461070000106X**[KL2]**Ibrahim Kirat and Ka-Sing Lau,*Classification of integral expanding matrices and self-affine tiles*, Discrete Comput. Geom.**28**(2002), no. 1, 49–73. MR**1904010**, 10.1007/s00454-001-0091-2**[Ko]**Mihail N. Kolountzakis,*The study of translational tiling with Fourier analysis*, Fourier analysis and convexity, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2004, pp. 131–187. MR**2087242****[KM]**Mihail N. Kolountzakis and Máté Matolcsi,*Complex Hadamard matrices and the spectral set conjecture*, Collect. Math.**Vol. Extra**(2006), 281–291. MR**2264214****[L]**I. Łaba,*The spectral set conjecture and multiplicative properties of roots of polynomials*, J. London Math. Soc. (2)**65**(2002), no. 3, 661–671. MR**1895739**, 10.1112/S0024610702003149**[LaW]**Izabella Łaba and Yang Wang,*On spectral Cantor measures*, J. Funct. Anal.**193**(2002), no. 2, 409–420. MR**1929508**, 10.1006/jfan.2001.3941**[LL]**King-Shun Leung and Ka-Sing Lau,*Disklikeness of planar self-affine tiles*, Trans. Amer. Math. Soc.**359**(2007), no. 7, 3337–3355. MR**2299458**, 10.1090/S0002-9947-07-04106-2**[LLR]**C.K. Lai, K.S. Lau and H. Rao,*Classification of the digit sets of self-similar tiles*, preprint.**[LR]**Ka-Sing Lau and Hui Rao,*On one-dimensional self-similar tilings and 𝑝𝑞-tiles*, Trans. Amer. Math. Soc.**355**(2003), no. 4, 1401–1414 (electronic). MR**1946397**, 10.1090/S0002-9947-02-03207-5**[LW1]**Jeffrey C. Lagarias and Yang Wang,*Tiling the line with translates of one tile*, Invent. Math.**124**(1996), no. 1-3, 341–365. MR**1369421**, 10.1007/s002220050056**[LW2]**Jeffrey C. Lagarias and Yang Wang,*Self-affine tiles in 𝑅ⁿ*, Adv. Math.**121**(1996), no. 1, 21–49. MR**1399601**, 10.1006/aima.1996.0045**[LW3]**Jeffrey C. Lagarias and Yang Wang,*Integral self-affine tiles in 𝐑ⁿ. I. Standard and nonstandard digit sets*, J. London Math. Soc. (2)**54**(1996), no. 1, 161–179. MR**1395075**, 10.1112/jlms/54.1.161**[LW4]**Jeffrey C. Lagarias and Yang Wang,*Integral self-affine tiles in 𝑅ⁿ. II. Lattice tilings*, J. Fourier Anal. Appl.**3**(1997), no. 1, 83–102. MR**1428817**, 10.1007/s00041-001-4051-2**[O]**A. M. Odlyzko,*Nonnegative digit sets in positional number systems*, Proc. London Math. Soc. (3)**37**(1978), no. 2, 213–229. MR**507604**, 10.1112/plms/s3-37.2.213**[P]**Vladimir Protasov,*Refinement equations with nonnegative coefficients*, J. Fourier Anal. Appl.**6**(2000), no. 1, 55–78. MR**1756136**, 10.1007/BF02510118**[SW]**Robert S. Strichartz and Yang Wang,*Geometry of self-affine tiles. I*, Indiana Univ. Math. J.**48**(1999), no. 1, 1–23. MR**1722192**, 10.1512/iumj.1999.48.1616**[T]**Terence Tao,*Fuglede’s conjecture is false in 5 and higher dimensions*, Math. Res. Lett.**11**(2004), no. 2-3, 251–258. MR**2067470**, 10.4310/MRL.2004.v11.n2.a8**[Th]**W. Thurston,*Groups, tilings and finite state automata*, in: AMS Colloquium Lecture Notes, 1989.**[V]**Andrew Vince,*Digit tiling of Euclidean space*, Directions in mathematical quasicrystals, CRM Monogr. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2000, pp. 329–370. MR**1798999**

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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

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