On one-dimensional self-similar tilings and -tiles

Authors:
Ka-Sing Lau and Hui Rao

Journal:
Trans. Amer. Math. Soc. **355** (2003), 1401-1414

MSC (2000):
Primary 52C20, 52C22; Secondary 42B99

Published electronically:
November 20, 2002

MathSciNet review:
1946397

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an integer base, a digit set and the set of radix expansions. It is well known that if has nonvoid interior, then can tile with some translation set ( is called a *tile* and a *tile digit set*). There are two fundamental questions studied in the literature: (i) describe the structure of ; (ii) for a given , characterize so that is a tile.

We show that for a given pair , there is a unique self-replicating translation set , and it has period for some . This completes some earlier work of Kenyon. Our main result for (ii) is to characterize the tile digit sets for when are distinct primes. The only other known characterization is for , due to Lagarias and Wang. The proof for the case depends on the techniques of Kenyon and De Bruijn on the cyclotomic polynomials, and also on an extension of the *product-form digit set* of Odlyzko.

**[B]**Christoph Bandt,*Self-similar sets. III. Constructions with sofic systems*, Monatsh. Math.**108**(1989), no. 2-3, 89–102. MR**1026611**, 10.1007/BF01308664**[BW]**C. Bandt and Y. Wang,*Disk-like self-affine tiles in ℝ²*, Discrete Comput. Geom.**26**(2001), no. 4, 591–601. MR**1863811**, 10.1007/s00454-001-0034-y**[DB]**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****[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**[HLR]**X. G. He, K. S. Lau and H. Rao, Self-affine sets and graph-directed systems,*Constr. Approx.*(to appear).**[KL]**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**[KLR]**I. Kirat, K. S. Lau and H. Rao, On the expanding polynomials and connectedness of self-affine tiles,*preprint*.**[K1]**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**[K2]**Richard Kenyon,*Projecting the one-dimensional Sierpinski gasket*, Israel J. Math.**97**(1997), 221–238. MR**1441250**, 10.1007/BF02774038**[LW1]**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**[LW2]**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**[LW3]**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**[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

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Additional Information

**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 and Nonlinear Science Center, Wuhan University, Wuhan, 430072, P.R. China;
Department of Mathematics, The Chinese University of Hong Kong, Hong Kong

Email:
raohui@tsuda.ac.jp

DOI:
http://dx.doi.org/10.1090/S0002-9947-02-03207-5

Received by editor(s):
February 13, 2002

Received by editor(s) in revised form:
September 11, 2002

Published electronically:
November 20, 2002

Additional Notes:
The authors are partially supported by an HKRGC grant and also a direct grant from CUHK. The second author is supported by CNSF 19901025.

Article copyright:
© Copyright 2002
American Mathematical Society