On beta expansions for Pisot numbers. (English) Zbl 0855.11039
If \(\beta >1\) is real then it is called a beta-number if the orbit of 1 under the map \(T(x):= \beta x\bmod 1\) is finite, in which case the sequence \(T^n (1)\) is ultimately periodic. Let \(p> 0\) be its period and let \(m\geq 0\) be the length of the preperiod. Since one has \(T^n (1)= P_n (\beta)\) with \(P_n (X)\in Z[ X]\), \(\beta\) is the root of \(R(X)\in Z[ X]\) defined by \(R(X)= P_{m+ p} (X)- P_m (X)\) if \(m> 0\) and \(R(X)= P_p (X)\) otherwise. Let \(P(X)\) be the minimal polynomial of \(\beta\). The author studies the polynomial \(Q(X)= R(X)/ P(X)\) and i.a. answers a question of I. Katai and C. Frougny by exhibiting infinitely many Pisot numbers \(\beta\) for which \(Q(X)\) is not a product of cyclotomic polynomials.
Reviewer: W.Narkiewicz (Wrocław)
MSC:
| 11K16 | Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. |
| 11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
| 11Y99 | Computational number theory |
Software:
MapleReferences:
| [1] | Mohamed Amara, Ensembles fermés de nombres algébriques, Ann. Sci. École Norm. Sup. (3) 83 (1966), 215 – 270 (1967) (French). · Zbl 0219.12004 |
| [2] | Daniel Berend and Christiane Frougny, Computability by finite automata and Pisot bases, Math. Systems Theory 27 (1994), no. 3, 275 – 282. · Zbl 0819.11005 · doi:10.1007/BF01578846 |
| [3] | M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, and J.-P. Schreiber, Pisot and Salem numbers, Birkhäuser Verlag, Basel, 1992. With a preface by David W. Boyd. · Zbl 0772.11041 |
| [4] | Anne Bertrand, Développements en base de Pisot et répartition modulo 1, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 6, A419 – A421 (French, with English summary). · Zbl 0362.10040 |
| [5] | Anne Bertrand-Mathis, Développement en base \?; répartition modulo un de la suite (\?\?\(^{n}\))_{\?\ge 0}; langages codés et \?-shift, Bull. Soc. Math. France 114 (1986), no. 3, 271 – 323 (French, with English summary). · Zbl 0628.58024 |
| [6] | David W. Boyd, Small Salem numbers, Duke Math. J. 44 (1977), no. 2, 315 – 328. · Zbl 0353.12003 |
| [7] | David W. Boyd, Pisot and Salem numbers in intervals of the real line, Math. Comp. 32 (1978), no. 144, 1244 – 1260. · Zbl 0395.12004 |
| [8] | David W. Boyd, Reciprocal polynomials having small measure, Math. Comp. 35 (1980), no. 152, 1361 – 1377. · Zbl 0447.12002 |
| [9] | David W. Boyd, Speculations concerning the range of Mahler’s measure, Canad. Math. Bull. 24 (1981), no. 4, 453 – 469. · Zbl 0474.12005 · doi:10.4153/CMB-1981-069-5 |
| [10] | David W. Boyd, Pisot numbers in the neighbourhood of a limit point. I, J. Number Theory 21 (1985), no. 1, 17 – 43. , https://doi.org/10.1016/0022-314X(85)90010-1 David W. Boyd, Pisot numbers in the neighborhood of a limit point. II, Math. Comp. 43 (1984), no. 168, 593 – 602. · Zbl 0575.12001 |
| [11] | David W. Boyd, Pisot numbers in the neighbourhood of a limit point. I, J. Number Theory 21 (1985), no. 1, 17 – 43. , https://doi.org/10.1016/0022-314X(85)90010-1 David W. Boyd, Pisot numbers in the neighborhood of a limit point. II, Math. Comp. 43 (1984), no. 168, 593 – 602. · Zbl 0575.12001 |
| [12] | David W. Boyd, Salem numbers of degree four have periodic expansions, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 57 – 64. |
| [13] | R. J. Bradford and J. H. Davenport, Effective tests for cyclotomic polynomials, Symbolic and algebraic computation (Rome, 1988) Lecture Notes in Comput. Sci., vol. 358, Springer, Berlin, 1989, pp. 244 – 251. · doi:10.1007/3-540-51084-2_22 |
| [14] | B.W. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monagan, and S.M. Watt, Maple V Language Reference Manual, Springer-Verlag, Berlin and New York, 1991. · Zbl 0758.68038 |
| [15] | J. Dufresnoy and Ch. Pisot, Étude de certaines fonctions méromorphes bornées sur le cercle unité, application à un ensemble fermé d’entiers algébriques, Ann. Sci. École Norm. Sup. (3) 72 (1955), 69–92. · Zbl 0064.03703 |
| [16] | ——, Sur les éléments d’accumulation d’un ensemble fermé d’entiers algébriques, Bull. Sci. Math. (2) 79 (1955), 54–64. · Zbl 0064.27303 |
| [17] | Leopold Flatto, Jeffrey C. Lagarias, and Bjorn Poonen, The zeta function of the beta transformation, Ergodic Theory Dynam. Systems 14 (1994), no. 2, 237 – 266. · Zbl 0843.58106 · doi:10.1017/S0143385700007860 |
| [18] | Christiane Frougny and Boris Solomyak, Finite beta-expansions, Ergodic Theory Dynam. Systems 12 (1992), no. 4, 713 – 723. · Zbl 0814.68065 · doi:10.1017/S0143385700007057 |
| [19] | A. O. Gel\(^{\prime}\)fond, A common property of number systems, Izv. Akad. Nauk SSSR. Ser. Mat. 23 (1959), 809 – 814 (Russian). |
| [20] | Wayne M. Lawton, A problem of Boyd concerning geometric means of polynomials, J. Number Theory 16 (1983), no. 3, 356 – 362. · Zbl 0516.12018 · doi:10.1016/0022-314X(83)90063-X |
| [21] | W. Parry, On the \?-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401 – 416 (English, with Russian summary). · Zbl 0099.28103 · doi:10.1007/BF02020954 |
| [22] | R. Salem, A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan, Duke Math. J. 11 (1944), 103–108. · Zbl 0063.06657 |
| [23] | Klaus Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), no. 4, 269 – 278. · Zbl 0494.10040 · doi:10.1112/blms/12.4.269 |
| [24] | Boris Solomyak, Conjugates of beta-numbers and the zero-free domain for a class of analytic functions, Proc. London Math. Soc. (3) 68 (1994), no. 3, 477 – 498. · Zbl 0820.30007 · doi:10.1112/plms/s3-68.3.477 |
| [25] | Faouzia Talmoudi, Sur les nombres de \?\cap [1,2], C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 16, A969 – A971 (French, with English summary). · Zbl 0377.12001 |
| [26] | Faouzia Lazami, Sur les éléments de \?\cap [1,2[, Séminaire Delange-Pisot-Poitou, 20e année: 1978/1979. Théorie des nombres, Fasc. 1 (French), Secrétariat Math., Paris, 1980, pp. Exp. No. 3, 6 (French). · Zbl 0411.12002 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
