On onedimensional selfsimilar tilings and tiles
Authors:
KaSing Lau and Hui Rao
Journal:
Trans. Amer. Math. Soc. 355 (2003), 14011414
MSC (2000):
Primary 52C20, 52C22; Secondary 42B99
Published electronically:
November 20, 2002
MathSciNet review:
1946397
Fulltext PDF Free Access
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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 selfreplicating 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 productform digit set of Odlyzko.
 [B]
Christoph
Bandt, Selfsimilar sets. III. Constructions with sofic
systems, Monatsh. Math. 108 (1989), no. 23,
89–102. MR
1026611 (91m:58050), http://dx.doi.org/10.1007/BF01308664
 [BW]
C.
Bandt and Y.
Wang, Disklike selfaffine tiles in ℝ², Discrete
Comput. Geom. 26 (2001), no. 4, 591–601. MR 1863811
(2002h:52028), http://dx.doi.org/10.1007/s004540010034y
 [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
(15,503b)
 [GH]
Karlheinz
Gröchenig and Andrew
Haas, Selfsimilar lattice tilings, J. Fourier Anal. Appl.
1 (1994), no. 2, 131–170. MR 1348740
(96j:52037), http://dx.doi.org/10.1007/s0004100140076
 [H]
John
E. Hutchinson, Fractals and selfsimilarity, Indiana Univ.
Math. J. 30 (1981), no. 5, 713–747. MR 625600
(82h:49026), http://dx.doi.org/10.1512/iumj.1981.30.30055
 [HLR]
X. G. He, K. S. Lau and H. Rao, Selfaffine sets and graphdirected systems, Constr. Approx. (to appear).
 [KL]
Ibrahim
Kirat and KaSing
Lau, On the connectedness of selfaffine tiles, J. London
Math. Soc. (2) 62 (2000), no. 1, 291–304. MR 1772188
(2001i:52027), http://dx.doi.org/10.1112/S002461070000106X
 [KLR]
I. Kirat, K. S. Lau and H. Rao, On the expanding polynomials and connectedness of selfaffine tiles, preprint.
 [K1]
Richard
Kenyon, Selfreplicating 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
(94a:52043), http://dx.doi.org/10.1090/conm/135/1185093
 [K2]
Richard
Kenyon, Projecting the onedimensional Sierpinski gasket,
Israel J. Math. 97 (1997), 221–238. MR 1441250
(98i:28002), http://dx.doi.org/10.1007/BF02774038
 [LW1]
Jeffrey
C. Lagarias and Yang
Wang, Selfaffine tiles in 𝑅ⁿ, Adv. Math.
121 (1996), no. 1, 21–49. MR 1399601
(97d:52034), http://dx.doi.org/10.1006/aima.1996.0045
 [LW2]
Jeffrey
C. Lagarias and Yang
Wang, Integral selfaffine tiles in 𝐑ⁿ. I. Standard
and nonstandard digit sets, J. London Math. Soc. (2)
54 (1996), no. 1, 161–179. MR 1395075
(97f:52031), http://dx.doi.org/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. 13, 341–365. MR 1369421
(96i:05040), http://dx.doi.org/10.1007/s002220050056
 [LW4]
Jeffrey
C. Lagarias and Yang
Wang, Integral selfaffine tiles in 𝑅ⁿ. II. Lattice
tilings, J. Fourier Anal. Appl. 3 (1997), no. 1,
83–102. MR
1428817 (98b:52026), http://dx.doi.org/10.1007/s0004100140512
 [O]
A.
M. Odlyzko, Nonnegative digit sets in positional number
systems, Proc. London Math. Soc. (3) 37 (1978),
no. 2, 213–229. MR 507604
(80m:10004), http://dx.doi.org/10.1112/plms/s337.2.213
 [B]
 C. Bandt, Selfsimilar sets 5. Integer matrices and fractal tilings of , Proc. Amer. Math. Soc. 112 (1991), 549562. MR 91m:58050
 [BW]
 C. Bandt and Y. Wang, Disklike selfaffine tiles in , Discrete Comput. Geom., 26 (2001), 591601. MR 2002h:52028
 [DB]
 N. G. De Bruijn, On the factorization of cyclic groups, Indag. Math. Kon. Akad. Wet., 15 (1953), 370377. MR 15:503b
 [GH]
 K. Gröchenig and A. Haas, Selfsimilar lattice tilings, J. Fourier Anal. Appl. 1 (1994), 131170. MR 96j:52037
 [H]
 J. E. Hutchinson, Fractals and selfsimilarity, Indiana Univ. Math. J. 30 (1981), 713747. MR 82h:49026
 [HLR]
 X. G. He, K. S. Lau and H. Rao, Selfaffine sets and graphdirected systems, Constr. Approx. (to appear).
 [KL]
 I. Kirat and K. S. Lau, On the connectedness of selfaffine tiles, J. London Math. Soc., 62 (2000), 291304. MR 2001i:52027
 [KLR]
 I. Kirat, K. S. Lau and H. Rao, On the expanding polynomials and connectedness of selfaffine tiles, preprint.
 [K1]
 R. Kenyon, Selfreplicating tilings, in Symbolic dynamics and its applications, Contemporary mathematics series, (P. Walters, ed.), American Mathematical Society, Providence, RI, vol. 135, 1992, pp. 239263. MR 94a:52043
 [K2]
 R. Kenyon, Projecting the onedimensional Sierpinski gasket, Israel J. Math, 97 (1997), 221238. MR 98i:28002
 [LW1]
 J. C. Lagarias and Y. Wang, Selfaffine tiles in , Adv. Math., 121 (1996), 2149. MR 97d:52034
 [LW2]
 J. C. Lagarias and Y. Wang, Integral selfaffine tiles in I. Standard and nonstandard digits sets, J. London Math. Soc., 54 (1996), 161179. MR 97f:52031
 [LW3]
 J. C. Lagarias and Y. Wang, Tiling the line with translation of one tile, Invent. Math., 124 (1996), 341365. MR 96i:05040
 [LW4]
 J. C. Lagarias and Y. Wang, Integral selfaffine tiles in II. Lattice tilings, J. Fourier Anal. Appl. 3 (1997), 84102. MR 98b:52026
 [O]
 A. M. Odlyzko, Nonnegative digit sets in positional number systems, Proc. London Math. Soc., 37 (1978), 213229. MR 80m:10004
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Additional Information
KaSing 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/S0002994702032075
PII:
S 00029947(02)032075
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
