Spectral structure of digit sets of self-similar tiles on ${\mathbb R}^1$
HTML articles powered by AMS MathViewer
- by Chun-Kit Lai, Ka-Sing Lau and Hui Rao PDF
- Trans. Amer. Math. Soc. 365 (2013), 3831-3850 Request permission
Abstract:
We study the structure of the digit sets ${\mathcal D}$ for the integral self-similar tiles $T(b,{\mathcal {D}})$ (we call such a ${\mathcal D}$ a tile digit set with respect to $b$). 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 $P_{\mathcal D}$, i.e., the zeros of $P_{\mathcal D}$ 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.References
- 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, DOI 10.1007/s10711-004-1774-7
- Christoph Bandt, Self-similar sets. V. Integer matrices and fractal tilings of $\textbf {R}^n$, Proc. Amer. Math. Soc. 112 (1991), no. 2, 549–562. MR 1036982, DOI 10.1090/S0002-9939-1991-1036982-1
- 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, DOI 10.1006/jabr.1998.7628
- 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
- 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, DOI 10.1016/j.jfa.2007.02.005
- Karlheinz Gröchenig and Andrew Haas, Self-similar lattice tilings, J. Fourier Anal. Appl. 1 (1994), no. 2, 131–170. MR 1348740, DOI 10.1007/s00041-001-4007-6
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- 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, DOI 10.1016/j.acha.2004.03.001
- 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, DOI 10.1007/s00365-002-0515-0
- 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, DOI 10.1090/conm/135/1185093
- 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, DOI 10.1112/S002461070000106X
- 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, DOI 10.1007/s00454-001-0091-2
- 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
- Mihail N. Kolountzakis and Máté Matolcsi, Complex Hadamard matrices and the spectral set conjecture, Collect. Math. Vol. Extra (2006), 281–291. MR 2264214
- 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, DOI 10.1112/S0024610702003149
- Izabella Łaba and Yang Wang, On spectral Cantor measures, J. Funct. Anal. 193 (2002), no. 2, 409–420. MR 1929508, DOI 10.1006/jfan.2001.3941
- 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, DOI 10.1090/S0002-9947-07-04106-2
- C.K. Lai, K.S. Lau and H. Rao, Classification of the digit sets of self-similar tiles, preprint.
- 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. MR 1946397, DOI 10.1090/S0002-9947-02-03207-5
- 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, DOI 10.1007/s002220050056
- Jeffrey C. Lagarias and Yang Wang, Self-affine tiles in $\textbf {R}^n$, Adv. Math. 121 (1996), no. 1, 21–49. MR 1399601, DOI 10.1006/aima.1996.0045
- Jeffrey C. Lagarias and Yang Wang, Integral self-affine tiles in $\mathbf R^n$. I. Standard and nonstandard digit sets, J. London Math. Soc. (2) 54 (1996), no. 1, 161–179. MR 1395075, DOI 10.1112/jlms/54.1.161
- Jeffrey C. Lagarias and Yang Wang, Integral self-affine tiles in $\textbf {R}^n$. II. Lattice tilings, J. Fourier Anal. Appl. 3 (1997), no. 1, 83–102. MR 1428817, DOI 10.1007/s00041-001-4051-2
- A. M. Odlyzko, Nonnegative digit sets in positional number systems, Proc. London Math. Soc. (3) 37 (1978), no. 2, 213–229. MR 507604, DOI 10.1112/plms/s3-37.2.213
- Vladimir Protasov, Refinement equations with nonnegative coefficients, J. Fourier Anal. Appl. 6 (2000), no. 1, 55–78. MR 1756136, DOI 10.1007/BF02510118
- Robert S. Strichartz and Yang Wang, Geometry of self-affine tiles. I, Indiana Univ. Math. J. 48 (1999), no. 1, 1–23. MR 1722192, DOI 10.1512/iumj.1999.48.1616
- Terence Tao, Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), no. 2-3, 251–258. MR 2067470, DOI 10.4310/MRL.2004.v11.n2.a8
- W. Thurston, Groups, tilings and finite state automata, in: AMS Colloquium Lecture Notes, 1989.
- 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, DOI 10.1112/s0024610700008711
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
- MR Author ID: 950029
- 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
- MR Author ID: 190087
- 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
- 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). - © Copyright 2013 American Mathematical Society
- 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
- MathSciNet review: 3042605