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2013 | 11 | 9 | 1616-1627
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Comments on the height reducing property

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A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring ℤ of the rational integers such that ℤ[α] = F[α]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. Expecting the converse of the last statement is true, we show some theoretical and experimental results, which support this conjecture.
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Bibliografia
  • [1] Akiyama S., Borbély T., Brunotte H., Peth"o A., Thuswaldner J.M., Generalized radix representations and dynamical systems. I, Acta Math. Hungar., 2005, 108(3), 207–238 http://dx.doi.org/10.1007/s10474-005-0221-z
  • [2] Akiyama S., Drungilas P., Jankauskas J., Height reducing problem on algebraic integers, Funct. Approx. Comment. Math., 2012, 47(1), 105–119 http://dx.doi.org/10.7169/facm/2012.47.1.9
  • [3] Akiyama S., Scheicher K., From number systems to shift radix systems, Nihonkai Math. J., 2005, 16(2), 95–106
  • [4] Baker A., The theory of linear forms in logarithms, In: Transcendence Theory: Advances and Applications, Cambridge, January–February, 1976, Academic Press, London, 1977, 1–27
  • [5] Baker A., Wüstholz G., Logarithmic forms and group varieties, J. Reine Angew. Math., 1993, 442, 19–62
  • [6] Dubickas A., Roots of polynomials with dominant term, Int. J. Number Theory, 2001, 7(5), 1217–1228 http://dx.doi.org/10.1142/S1793042111004575
  • [7] Frougny C., Steiner W., Minimal weight expansions in Pisot bases, J. Math. Cryptol., 2008, 2(4), 365–392 http://dx.doi.org/10.1515/JMC.2008.017
  • [8] Lagarias J.C., Wang Y., Integral self-affine tiles in ℝn. Part II: Lattice tilings, J. Fourier Anal. Appl., 1997, 3(1), 83–102 http://dx.doi.org/10.1007/BF02647948
  • [9] Lenstra A.K., Lenstra H.W. Jr., Lovász L., Factoring polynomials with rational coefficients, Math. Ann., 1982, 261(4), 515–534 http://dx.doi.org/10.1007/BF01457454
  • [10] Mahler K., A remark on Kronecker’s theorem, Enseignement Math., 1966, 12, 183–189
  • [11] Narkiewicz W., Elementary and Analytic Theory of Algebraic Numbers, 3rd ed., Springer Monogr. Math., Springer, Berlin, 2004 http://dx.doi.org/10.1007/978-3-662-07001-7
  • [12] Vorselen T., On Kronecker’s Theorem over the Adéles, MSc thesis, Universiteit Leiden, 2010, preprint available at http://www.math.leidenuniv.nl/scripties/VorselenMaster.pdf
  • [13] Waldschmidt M., A lower bound for linear forms in logarithms, Acta Arith., 1980, 37, 257–283
  • [14] de Weger B.M.M., Algorithms for Diophantine Equations, CWI Tract, 65, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989
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bwmeta1.element.doi-10_2478_s11533-013-0262-4
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