Adamczewski, Boris; Bugeaud, Yann Dynamics for \(\beta \)-shifts and Diophantine approximation. (English) Zbl 1140.11035 Ergodic Theory Dyn. Syst. 27, No. 6, 1695-1711 (2007). The authors investigate the \(\beta \)-expansion of an algebraic number in an algebraic base \(\beta \). Using tools from Diophantine approximation, they prove several results that may suggest a strong difference between the asymptotic behaviour of eventually periodic expansions and that of non-eventually periodic expansions. Among interesting results proved by the authors, we give the following one. Let \(\beta\) be a Pisot or a Salem number. Let \(\alpha\) be an algebraic number in \([0,1)\). Then the \(\beta\)-expansion of \(\alpha\) can be generated by a finite automaton if and only if it is eventually periodic. Reviewer: Jean-Paul Allouche (Orsay) Cited in 2 ReviewsCited in 23 Documents MSC: 11J81 Transcendence (general theory) 11J13 Simultaneous homogeneous approximation, linear forms 37B10 Symbolic dynamics 68R15 Combinatorics on words 11B85 Automata sequences Keywords:diophantine approximation; \(\beta\)-shift; transcendence; Pisot and Salem numbers; linear forms with algebraic coefficients PDFBibTeX XMLCite \textit{B. Adamczewski} and \textit{Y. Bugeaud}, Ergodic Theory Dyn. Syst. 27, No. 6, 1695--1711 (2007; Zbl 1140.11035) Full Text: DOI References: [1] Verger-Gaugry, Ann. Inst. Fourier (Grenoble) 56 pp 2565– (2006) · Zbl 1177.11013 [2] DOI: 10.1017/S0143385797079182 · Zbl 0908.58017 [3] DOI: 10.1007/BF02020954 · Zbl 0099.28103 [4] DOI: 10.1007/BF01534862 [5] Bertrand-Mathis, Bull. Soc. Math. France 114 pp 271– (1986) [6] DOI: 10.1007/BF02588048 · Zbl 1195.11093 [7] Borel, C. R. Acad. Sci. Paris 230 pp 591– (1950) [8] DOI: 10.1007/BF02020331 · Zbl 0079.08901 [9] DOI: 10.1016/0304-3975(89)90038-8 · Zbl 0682.68081 [10] Gazeau, J. Th?or. Nombres Bordeaux 16 pp 125– (2004) · Zbl 1075.11007 [11] Adamczewski, Ann. of Math. (2) 165 pp 547– (2007) · Zbl 1195.11094 [12] DOI: 10.1016/j.tcs.2004.03.035 · Zbl 1068.68112 [13] DOI: 10.1007/BF01454845 · JFM 55.0115.01 [14] DOI: 10.2307/2695302 · Zbl 0997.11052 [15] Ito, J. Math. Soc. Japan 27 pp 20– (1975) · Zbl 0292.10040 [16] DOI: 10.1023/A:1015594913393 · Zbl 1010.11038 [17] Evertse, Compos. Math. 101 pp 225– (1996) [18] Adamczewski, C. R. Acad. Sci. Paris 339 pp 11– (2004) · Zbl 1119.11019 [19] DOI: 10.1016/j.aam.2006.04.001 · Zbl 1117.68059 [20] Berth?, Integers 5 pp none– (2005) [21] Bertrand-Mathis, Ergod. Th. and Dynam. Sys. 8 pp 35– (1988) · Zbl 0657.28014 [22] Lagarias, Experiment. Math. 10 pp 355– (2001) · Zbl 1015.11036 [23] DOI: 10.1090/S0025-5718-96-00700-4 · Zbl 0848.11048 [24] DOI: 10.1112/blms/12.4.269 · Zbl 0494.10040 [25] Bailey, Experiment. Math. 10 pp 175– (2001) · Zbl 1047.11073 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.