zbMATH — the first resource for mathematics

Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions. (English) Zbl 1190.11005
This paper under review studies tilings and representation spaces related to the \(\beta\)-transformation when \(\beta\) is a Pisot number, that is not a unit. The obtained results are applied to study the set of rational numbers having a purely periodic \(\beta\)-expansion. The authors make use of the connection between pure periodicity and a compact self-similar representation of number having no fractional part in their \(\beta\)-expansion, called center tile. For elements \(x\) of the ring \(\mathbb Z[1/\beta]\), so-called \(x\)-tiles are introduced, so that the central tile is a finite union of \(x\)-tiles up to translation. These \(x\)-tiles provide a covering (and even in some cases a tiling) of the space the authors working in. This space, called complete representation space, is based on archimedean as well as on the non-archimedean completions of the number field \(\mathbb Q(b)\) corresponding to the prime divisors of the norm of \(\beta\). This representation space has numerous potential implications. The authors focus on the gamma function \(\gamma(\beta)\) defined as the supremum of the set of elements \(v\) in \([0,1]\) such that every positive rational number \(p/q\), with \(p/q\leq v\) and \(q\) coprime with the norm of \(\beta\), has a purely periodic \(\beta\)-expansion. The key point relies on the description of the boundary of the tiles in terms of paths on a graph called “boundary graph”. The paper ends with explicit quadratic examples, showing that the general behaviour of \(\gamma(\beta)\) is slightly more complicated than in the unit case.

11A63 Radix representation; digital problems
03D45 Theory of numerations, effectively presented structures
11S99 Algebraic number theory: local fields
28A75 Length, area, volume, other geometric measure theory
52C23 Quasicrystals and aperiodic tilings in discrete geometry
Full Text: DOI
[1] Akiyama, S.: Pisot Numbers and Greedy Algorithm. Number Theory (Eger, 1996), pp. 9–21. de Gruyter, Berlin (1998) · Zbl 0919.11063
[2] Akiyama, S.: Cubic Pisot Units With Finite Beta Expansions. Algebraic Number Theory and Diophantine Analysis (Graz, 1998), pp. 11–26. de Gruyter, Berlin (2000) · Zbl 1001.11038
[3] Akiyama S.: On the boundary of self affine tilings generated by Pisot numbers. J. Math. Soc. Jpn 54(2), 283–308 (2002) · Zbl 1032.11033
[4] Akiyama, S.: Pisot Number System and its Dual Tiling. Physics and Theoretical Computer Science (Cargese, 2006), pp. 133–154. IOS Press, Amsterdam (2007)
[5] Akiyama S., Scheicher K.: Intersecting two dimensional fractals and lines. Acta Sci. Math. (Szeged) 3-4, 555–580 (2005) · Zbl 1111.11006
[6] Barat G., Berthé V., Liardet P., Thuswaldner J.M.: Dynamical directions in numeration. Ann. Inst. Fourier (Grenoble) 56(7), 1987–2092 (2006) · Zbl 1138.37005
[7] Bedford, T.: Applications of dynamical systems theory to fractals–a study of cookie-cutter Cantor sets. In: Fractal Geometry and Analysis (Montreal, PQ, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 346, pp. 1–44. Kluwer, Dordrecht (1991) MR MR1140719 · Zbl 0741.58011
[8] Berthé, V., Siegel, A.: Tilings associated with beta-numeration and substitutions. Integers 5(3), A2, 46 pp. (2005) (electronic) · Zbl 1139.37008
[9] Berthé V., Siegel A.: Purely periodic {\(\beta\)}-expansions in the Pisot non-unit case. J. Number Theor. 127, 153–172 (2007) · Zbl 1197.11139
[10] Bertrand A.: Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris Sér. A-B 285(6), A419–A421 (1977) · Zbl 0362.10040
[11] Blanchard F.: {\(\beta\)}-expansions and symbolic dynamics. Theoret. Comput. Sci. 65(2), 131–141 (1989) · Zbl 0682.68081
[12] Burdík Č, Frougny Ch., Gazeau J.P., Krejcar R.: Beta-integers as natural counting systems for quasicrystals. J. Phys. A 31(30), 6449–6472 (1998) · Zbl 0941.52019
[13] Cassels J.W.S., Fröhlich A.: Algebraic Number Theory. Academic Press, London (1986)
[14] Cornfeld, I.P., Fomin, S.V., Sinaĭ, Ya.G.: Ergodic Theory, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 245, Springer, New York (1982), Translated from the Russian by A. B. Sosinskiĭ. MR MR832433 (87f:28019) · Zbl 0493.28007
[15] Dajani K., Kraaikamp C., Solomyak B.: The natural extension of the {\(\beta\)}-transformation. Acta Math. Hungar. 73(1–2), 97–109 (1996) MR MR1415923 (99d:28029) · Zbl 0931.28014
[16] Frougny, C.: Number representation and finite automata. In: Topics in Symbolic Dynamics and Applications (Temuco, 1997), London Mathamaticlal Society Lecture Note Series, vol. 279, pp. 207–228. Cambridge University Press, Cambridge (2000) · Zbl 0976.11003
[17] Frougny C., Solomyak B.: Finite beta-expansions. Ergodic Theor. Dyn. Syst. 12, 45–82 (1992) · Zbl 0814.68065
[18] Hama M., Imahashi T.: Periodic {\(\beta\)}-expansions for certain classes of Pisot numbers. Comment. Math. Univ. St Paul. 46(2), 103–116 (1997) · Zbl 0899.11039
[19] Ito S., Rao H.: Purely periodic {\(\beta\)}-expansions with Pisot unit base. Proc. Am. Math. Soc. 133(4), 953–964 (2004) (electronic) · Zbl 1099.11062
[20] Lindenstrauss E., Schmidt K.: Symbolic representations of nonexpansive group automorphisms. Israel J. Math. 149, 227–266 (2005) · Zbl 1087.37010
[21] Lothaire, M.: Algebraic combinatorics on words. In: Encyclopedia of Mathematics and its Applications, vol. 90 (Chap. 7, written by C. Frougny), Cambridge University Press, Cambridge (2002) · Zbl 1001.68093
[22] Parry W.: On the {\(\beta\)}-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11, 401–416 (1960) · Zbl 0099.28103
[23] Praggastis B.: Numeration systems and Markov partitions from self-similar tilings. Trans. Am. Math. Soc. 351(8), 3315–3349 (1999) · Zbl 0984.11008
[24] Qu Y.-H., Rao H., Yang Y.-M.: Periods of {\(\beta\)}-expansions and linear recurrent sequences. Acta Arith. 120(1), 27–37 (2005) · Zbl 1155.11337
[25] Rauzy, G.: Rotations sur les groupes, nombres algébriques, et substitutions, Séminaire de Théorie des Nombres, 1987–1988 (Talence, 1987–1988), Univ. Bordeaux I, Talence, 1988, Exp. No. 21, 12. MR 90g:11017
[26] Rohlin V.A.: Exact endomorphisms of a Lebesgue space. Izv. Akad. Nauk SSSR Ser. Mat. 25, 499–530 (1961) MR MR0143873 (26 #1423)
[27] Sano Y.: On purely periodic beta-expansions of Pisot numbers. Nagoya Math. J. 166, 183–207 (2002) · Zbl 1029.11040
[28] Schmidt K.: On periodic expansions of Pisot numbers and Salem numbers. Bull. Lond. Math. Soc. 12(4), 269–278 (1980) · Zbl 0494.10040
[29] Schmidt K.: Algebraic coding of expansive group automorphisms and two-sided beta-shifts. Monatsh. Math. 129(1), 37–61 (2000) MR 2001f:54043 · Zbl 1010.37005
[30] Siegel A.: Représentation des systèmes dynamiques substitutifs non unimodulaires. Ergodic Theor. Dyn. Syst. 23(4), 1247–1273 (2003) · Zbl 1052.37009
[31] Siegel, A., Thuswaldner, J.M.: Topological properties of rauzy fractals, preprint, 2007 · Zbl 1229.28021
[32] Sing B.: Iterated function systems in mixed Euclidean and p-adic spaces. In: Hackensack N.J., ((eds) Complexus Mundi., pp. 267–276. World Science, Singapore (2006) MR MR2227210 · Zbl 1163.28307
[33] Sirvent V.F., Wang Y.: Self-affine tiling via substitution dynamical systems and Rauzy fractals. Pacific J. Math. 206(2), 465–485 (2002) MR 2003g:37026 · Zbl 1048.37015
[34] Thurston, W.P.: Groups, tilings and finite state automata, AMS Colloquium lectures, AMS Colloquium lectures, 1989 · Zbl 0713.26003
[35] Thuswaldner J.M.: Unimodular Pisot substitutions and their associated tiles. J. Théor. Nombres Bordeaux 18(2), 487–536 (2006) MR MR2289436 · Zbl 1161.37016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.