Karyon expansions of Pisot numbers in multidimensional continued fractions

V. G. Zhuravlev

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 32, Tome 449 (2016), pp. 168-195

Résumé

Best simultaneously diophantine a pproximations are obtained for Pisot numbers. For this purpose the karyon approximation method is used. Littlewood–Pisot numbers are investigated in detail.

Bibliographie

[1] V. G. Zhuravlev, “Differentsirovanie indutsirovannykh razbienii tora i mnogomernye priblizheniya algebraicheskikh chisel”, Zap. nauchn. semin. POMI, 445, 2016, 33–92

[2] V. G. Zhuravlev, “Dvumernye priblizheniya metodom delyaschikhsya toricheskikh razbienii”, Zap. nauchn. semin. POMI, 440, 2015, 81–98

[3] V. G. Zhuravlev, “Delyaschiesya razbieniya tora i mnozhestva ogranichennogo ostatka”, Zap. nauchn. semin. POMI, 440, 2015, 99–122

[4] V. G. Zhuravlev, “Periodicheskie yadernye razlozheniya kubicheskikh irratsionalnostei v tsepnye drobi”, Matematika i informatika, Sovr. probl. matem., M., 2016 (to appear) , 30 s. (MI RAN)

[5] Z. Coelho, A. Lopes, L. F. Da Rocha, “Absolutely Continuous Invariant Measures for a Class of Affine Interval Exchange Maps”, Proc. Amer. Math. Soc., 123:11 (1995), 3533–3542 | DOI | MR | Zbl

[6] V. G. Zhuravlev, A. V. Shutov, “Derivaties of circle rotations and similarity of orbits”, Max Planck Institut für Math., Preprint Series, 62 (2004), 1–11 | MR

[7] M. Furukado, Sh. Ito, A. Saito, J. Tamura, Sh. Yasutomi, “A new multidimensional slow continued fraction algorithm and stepped surface”, Exper. Math., 23:4 (2014), 390–410 | DOI | MR | Zbl

[8] M. Abrate, S. Barbero, U. Cerruti, N. Murru, “Periodic Representations for Cubic Irrationalities”, Fibonacci Quart., 50:3 (2012), 252–264 | MR | Zbl

[9] N. Murru, “On the periodic writing of cubic irrationals and a generalization of Rédei functions”, Int. J. Number Theory, 11 (2015), 779–799 | DOI | MR | Zbl

[10] Sh. Ito, J. Fujii, H. Higashino, Sh. Yasutomi, “On simultaneous approximation to $(\alpha,\alpha^2)$ with $\alpha^3+k\alpha-1=0$”, J. Number Theory, 99 (2003), 255–283 | DOI | MR | Zbl

[11] Q. Wang, K. Wang, Z. Dai, “On optimal simultaneous rational approximation to $(\omega,\omega^2)^\tau$ with $\omega$ being some kind of cubic algebraic function”, J. Approx. Theory, 148 (2007), 194–210 | DOI | MR | Zbl

[12] P. Hubert, A. Messaoudi, “Best simultaneous Diophantine approximations of Pisot numbers and Rauzy fractals”, Acta Arith., 124:1 (2006), 1–5 | DOI | MR

[13] N. Chevallier, “Best simultaneous Diophantine approximations of some cubic numbers”, Journal de Théories des Nombres de Bordeaux, 14:2 (2002), 403–414 | DOI | MR | Zbl

[14] N. Chevallier, “Best Simultaneous Diophantine Approximations and Multidimensional Continued Fraction Expansions”, Moscow J. Combinatorics and Number Theory, 3:1 (2013), 3–56 | MR | Zbl

[15] A. Ya. Khinchin, Tsepnye drobi, 4-oe izd., M., 1978

[16] V. G. Zhuravlev, “Periodicheskie yadernye razlozheniya edinits algebraicheskikh polei v tsepnye drobi”, Zap. nauchn. semin. POMI, 449, 2016, 84–129

[17] J. Lagarias, “Best simultaneous Diophantine approximations. I. Groth rates of best approximation denomimators”, Trans. Amer. Math. Soc., 272 (1982), 545–554 | MR | Zbl

[18] Dzh. Kassels, Vvedenie v teoriyu diofantovykh priblizhenii, M., 1961

[19] E. S. Fedorov, Nachala ucheniya o figurakh, M., 1953 | MR

[20] G. F. Voronoi, Sobranie sochinenii, v. 2, Kiev, 1952

[21] V. G. Zhuravlev, “Perekladyvayuschiesya toricheskie razvertki i mnozhestva ogranichennogo ostatka”, Zap. nauchn. semin. POMI, 392, 2011, 95–145

[22] V. G. Zhuravlev, “Mnogogranniki ogranichennogo ostatka”, Matematika i informatika, 1, Sovr. probl. matem., 16, MI RAN, M., 2012, 82–102 | DOI | Zbl

[23] K. Mukunda, “Littlewood Pisot numbers”, J. Number Theory, 117 (2006), 106–121 | DOI | MR | Zbl

[24] G. Rauzy, “Nombres algébriques et substitutions”, Bull. Soc. Math. France, 110 (1982), 147–178 | MR | Zbl

[25] V. G. Zhuravlev, “Razbieniya Rozi i mnozhestva ogranichennogo ostatka”, Zap. nauchn. semin. POMI, 322, 2005, 83–106 | MR | Zbl