Spectral structure of digit sets of selfsimilar tiles on
Authors:
ChunKit Lai, KaSing Lau and Hui Rao
Journal:
Trans. Amer. Math. Soc. 365 (2013), 38313850
MSC (2010):
Primary 11A63; Secondary 11B75, 28A80, 52C22
Published electronically:
February 26, 2013
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Abstract 
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Similar Articles 
Additional Information
Abstract: We study the structure of the digit sets for the integral selfsimilar 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 productforms. 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 selfsimilar 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 productform to higher order.
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S. Akiyama and J. Thuswaldner, A survey on topological properties of tiles related to number systems, Geom. Dedicata, vol. 109 (2004) 89105. MR 2113188 (2005h:37035)
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C. Bandt, Selfsimilar sets 5. Integer matrices and fractal tilings of , Proc. Amer. Math. Soc. 112(1991), 549562. MR 1036982 (92d:58093)
 [CM]
E. Coven and A. Meyerowitz, Tiling the integers with translates of one finite set, J. Algebra 212(1999), 161174. MR 1670646 (99k:11032)
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N. De Bruijn, On the factorization of cyclic groups, Indag. Math. Kon. Akad. Wet., 15 (1953), 370377. MR 0059271 (15:503b)
 [DFW]
X.R. Dai, D.J. Feng and Yang Wang, Refinable functions with noninteger dilations., J. Func. Anal. 250 (2007), 120. MR 2345903 (2008i:42069)
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K. Gröchenig and A. Haas, Selfsimilar lattice tilings, J. Fourier Anal. Appl. 1 (1994), 131170. MR 1348740 (96j:52037)
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J. Hutchinson, Fractals and selfsimilarity, Indiana Univ. Math. J., 30 (1981), 713  747. MR 625600 (82h:49026)
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X.G. He and K.S. Lau, Characterization of tile digit sets with prime determinants, Applied and Computational Harmonic Analysis, 16 (2004), 159  173. MR 2054276 (2005c:52019)
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X.G. He, K.S. Lau and H. Rao, Selfaffine sets and graphdirected systems, Constr. Appr., 19 (2003), 373397. MR 1979057 (2004k:52026)
 [K]
R. Kenyon, Selfreplicating tilings, Symbolic Dynamics and its Applications (ed. P. Walters), Amer. Math. Soc., Providence, RI, 1992, 239264. MR 1185093 (94a:52043)
 [KL1]
I. Kirat and K.S. Lau, On the connectedness of selfaffine tiles, J. London Math. Soc. 62 (2000), 291304. MR 1772188 (2001i:52027)
 [KL2]
I. Kirat and K.S. Lau, Classification of integral expanding matrices and selfaffine tiles, Discrete Comput. Geom. 28 (2002), 4973. MR 1904010 (2003c:52030)
 [Ko]
M. Kolounzakis, The study of translation tiling with Fourier analysis, Fourier Analysis and Convexity, Appl. Numer. Harmonic Anal., Birkhäuser, Boston, 2004, 131187. MR 2087242 (2005e:42071)
 [KM]
M. Kolountzakis and M. Matolcsi, Complex Hadamard matrices and the Spectral Set Conjecture, Collectanea Mathematica, Vol. Extra (2006), 281  291. MR 2264214 (2007h:52023)
 [L]
I. Łaba, The spectral set conjecture and multiplicative properties of roots of polynomials, J. London Math. Soc., 65 (2001), 661671. MR 1895739 (2003e:42015)
 [LaW]
I. Łaba and Y. Wang, On spectral Cantor measures, J. Funct. Anal., 193 (2002), 409  420. MR 1929508 (2003g:28017)
 [LL]
K.S. Leung and K.S. Lau, Disklikeness of planar selfaffine tiles, Trans. Amer. Math. Soc. 359 (2007), 33373355. MR 2299458 (2008k:52046)
 [LLR]
C.K. Lai, K.S. Lau and H. Rao, Classification of the digit sets of selfsimilar tiles, preprint.
 [LR]
K.S. Lau and H. Rao, On onedimensional selfsimilar tilings and the tilings, Trans. Amer. Math. Soc., 355 (2003), 1401  1414. MR 1946397 (2003k:11033)
 [LW1]
J. Lagarias and Y. Wang, Tiling the line by translates of one tile, Inventiones Math., 124 (1996), 341  365. MR 1369421 (96i:05040)
 [LW2]
J. Lagarias and Y. Wang, Selfaffine tiles in , Adv. in Math., 121 (1996), 21  49. MR 1399601 (97d:52034)
 [LW3]
J. Lagarias and Y. Wang, Integral selfaffine tiles in I. Standard and nonstandard digit sets, J. London Math. Soc., 53 (1996), 161  179. MR 1395075 (97f:52031)
 [LW4]
J. Lagarias and Y. Wang, Integral selfaffine tiles, Part II. Lattice tilings, J. Fourier Anal. Appl. 3 (1997), 84102. MR 1428817 (98b:52026)
 [O]
A. Odlyzko, Nonnegative digit sets in positional number systems, Proc. London Math. Soc. (3), 37 (1978), 213229. MR 507604 (80m:10004)
 [P]
V. Protasov, Refinement equations with nonnegative coefficients, J. Fourier Anal. Appl., 6 (2000), 5578. MR 1756136 (2001i:42008)
 [SW]
R. Strichartz and Y. Wang, Geometry of selfaffine tiles I, Indiana Univ. Math. J. 48 (1999), 123. MR 1722192 (2000k:52017)
 [T]
T. Tao, Fuglede's conjecture is false in and higher dimensions, Math. Res. Letter, 11 (2004), 251258. MR 2067470 (2005i:42037)
 [Th]
W. Thurston, Groups, tilings and finite state automata, in: AMS Colloquium Lecture Notes, 1989.
 [V]
A. Vince, Digit tiling of Euclidean space, in: Direction in Mathematical Quasicrystals, in: CRM Monograph Ser., vol. 13, AMS Providence, RI, 2000, 329370. MR 1798999 (2002g:52025)
 [AT]
 S. Akiyama and J. Thuswaldner, A survey on topological properties of tiles related to number systems, Geom. Dedicata, vol. 109 (2004) 89105. MR 2113188 (2005h:37035)
 [B]
 C. Bandt, Selfsimilar sets 5. Integer matrices and fractal tilings of , Proc. Amer. Math. Soc. 112(1991), 549562. MR 1036982 (92d:58093)
 [CM]
 E. Coven and A. Meyerowitz, Tiling the integers with translates of one finite set, J. Algebra 212(1999), 161174. MR 1670646 (99k:11032)
 [deB]
 N. De Bruijn, On the factorization of cyclic groups, Indag. Math. Kon. Akad. Wet., 15 (1953), 370377. MR 0059271 (15:503b)
 [DFW]
 X.R. Dai, D.J. Feng and Yang Wang, Refinable functions with noninteger dilations., J. Func. Anal. 250 (2007), 120. MR 2345903 (2008i:42069)
 [GH]
 K. Gröchenig and A. Haas, Selfsimilar lattice tilings, J. Fourier Anal. Appl. 1 (1994), 131170. MR 1348740 (96j:52037)
 [H]
 J. Hutchinson, Fractals and selfsimilarity, Indiana Univ. Math. J., 30 (1981), 713  747. MR 625600 (82h:49026)
 [HL]
 X.G. He and K.S. Lau, Characterization of tile digit sets with prime determinants, Applied and Computational Harmonic Analysis, 16 (2004), 159  173. MR 2054276 (2005c:52019)
 [HLR]
 X.G. He, K.S. Lau and H. Rao, Selfaffine sets and graphdirected systems, Constr. Appr., 19 (2003), 373397. MR 1979057 (2004k:52026)
 [K]
 R. Kenyon, Selfreplicating tilings, Symbolic Dynamics and its Applications (ed. P. Walters), Amer. Math. Soc., Providence, RI, 1992, 239264. MR 1185093 (94a:52043)
 [KL1]
 I. Kirat and K.S. Lau, On the connectedness of selfaffine tiles, J. London Math. Soc. 62 (2000), 291304. MR 1772188 (2001i:52027)
 [KL2]
 I. Kirat and K.S. Lau, Classification of integral expanding matrices and selfaffine tiles, Discrete Comput. Geom. 28 (2002), 4973. MR 1904010 (2003c:52030)
 [Ko]
 M. Kolounzakis, The study of translation tiling with Fourier analysis, Fourier Analysis and Convexity, Appl. Numer. Harmonic Anal., Birkhäuser, Boston, 2004, 131187. MR 2087242 (2005e:42071)
 [KM]
 M. Kolountzakis and M. Matolcsi, Complex Hadamard matrices and the Spectral Set Conjecture, Collectanea Mathematica, Vol. Extra (2006), 281  291. MR 2264214 (2007h:52023)
 [L]
 I. Łaba, The spectral set conjecture and multiplicative properties of roots of polynomials, J. London Math. Soc., 65 (2001), 661671. MR 1895739 (2003e:42015)
 [LaW]
 I. Łaba and Y. Wang, On spectral Cantor measures, J. Funct. Anal., 193 (2002), 409  420. MR 1929508 (2003g:28017)
 [LL]
 K.S. Leung and K.S. Lau, Disklikeness of planar selfaffine tiles, Trans. Amer. Math. Soc. 359 (2007), 33373355. MR 2299458 (2008k:52046)
 [LLR]
 C.K. Lai, K.S. Lau and H. Rao, Classification of the digit sets of selfsimilar tiles, preprint.
 [LR]
 K.S. Lau and H. Rao, On onedimensional selfsimilar tilings and the tilings, Trans. Amer. Math. Soc., 355 (2003), 1401  1414. MR 1946397 (2003k:11033)
 [LW1]
 J. Lagarias and Y. Wang, Tiling the line by translates of one tile, Inventiones Math., 124 (1996), 341  365. MR 1369421 (96i:05040)
 [LW2]
 J. Lagarias and Y. Wang, Selfaffine tiles in , Adv. in Math., 121 (1996), 21  49. MR 1399601 (97d:52034)
 [LW3]
 J. Lagarias and Y. Wang, Integral selfaffine tiles in I. Standard and nonstandard digit sets, J. London Math. Soc., 53 (1996), 161  179. MR 1395075 (97f:52031)
 [LW4]
 J. Lagarias and Y. Wang, Integral selfaffine tiles, Part II. Lattice tilings, J. Fourier Anal. Appl. 3 (1997), 84102. MR 1428817 (98b:52026)
 [O]
 A. Odlyzko, Nonnegative digit sets in positional number systems, Proc. London Math. Soc. (3), 37 (1978), 213229. MR 507604 (80m:10004)
 [P]
 V. Protasov, Refinement equations with nonnegative coefficients, J. Fourier Anal. Appl., 6 (2000), 5578. MR 1756136 (2001i:42008)
 [SW]
 R. Strichartz and Y. Wang, Geometry of selfaffine tiles I, Indiana Univ. Math. J. 48 (1999), 123. MR 1722192 (2000k:52017)
 [T]
 T. Tao, Fuglede's conjecture is false in and higher dimensions, Math. Res. Letter, 11 (2004), 251258. MR 2067470 (2005i:42037)
 [Th]
 W. Thurston, Groups, tilings and finite state automata, in: AMS Colloquium Lecture Notes, 1989.
 [V]
 A. Vince, Digit tiling of Euclidean space, in: Direction in Mathematical Quasicrystals, in: CRM Monograph Ser., vol. 13, AMS Providence, RI, 2000, 329370. MR 1798999 (2002g:52025)
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Additional Information
ChunKit 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
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, Central China Normal University, Wuhan, People’s Republic of China
Email:
hrao@mail.ccnu.edu.cn
DOI:
http://dx.doi.org/10.1090/S00029947201305787X
PII:
S 00029947(2013)05787X
Keywords:
Blocking,
cyclotomic polynomials,
kernel polynomials,
prime,
productforms,
selfsimilar 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
