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Mahler Measures Close to an Integer

Published online by Cambridge University Press:  20 November 2018

Artūras Dubickas
Artūras Dubickas*
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania, e-mail: arturas.dubickas@maf.vu.lt website: http://www.mif.vu.lt/ttsk/bylos/da/da_a.html
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Abstract

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We prove that the Mahler measure of an algebraic number cannot be too close to an integer, unless we have equality. The examples of certain Pisot numbers show that the respective inequality is sharp up to a constant. All cases when the measure is equal to the integer are described in terms of the minimal polynomials.

Keywords

11R0411R0611R0911J68Mahler measurePV numbersSalem numbers
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 45 , Issue 2 , 01 June 2002 , pp. 196 - 203
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Adler, R. L. and Marcus, B., Topological entropy and equivalence of dynamical systems. Mem. Amer. Math. Soc. (219) 20(1979).Google Scholar
[2] Boyd, D. W., Reciprocal polynomials having small measure.Math. Comp. 35 (1980), 35–1980.Google Scholar
[3] Boyd, D. W., Speculations concerning the range of the Mahler measure. Canad. Math. Bull. 24 (1981), 24–1981.Google Scholar
[4] Boyd, D. W., Perron units which are not Mahler measures. Ergodic Theory Dynamical Systems 6 (1986), 6–1986.Google Scholar
[5] Boyd, D. W., Reciprocal algebraic integers whose Mahler measures are non-reciprocal. Canad. Math. Bull. 30 (1987), 30–1987.Google Scholar
[6] Boyd, D. W., Reciprocal polynomials having small measure. II. Math. Comp. 53 (1989), 53–1989.Google Scholar
[7] Dobrowolski, E., On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith. 34 (1979), 34–1979.Google Scholar
[8] Dubickas, A., Algebraic conjugates outside the unit circle. In: New Trends in Probability and Statistics Vol. 4: Analytic and Probabilistic Methods in Number Theory, Palanga, 1996 (eds. A. Laurincikas et al.), TEV Vilnius, VSP Utrecht, 1997, 11–21.Google Scholar
[9] Dubickas, A., Polynomials with a root close to an integer. Liet. Matem. Rink. 39 (1999), 39–1999.Google Scholar
[10] Dubickas, A. and Smyth, C. J., On the Remak height, the Mahler measure, and conjugate sets of algebraic numbers lying on two circles. Proc. Edinburgh Math. Soc., to appear.Google Scholar
[11] Everest, G. and Ward, T., Heights of polynomials and entropy in algebraic dynamics. Springer, London, 1999.Google Scholar
[12] Kronecker, L., Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten. J. Reine Angew. Math. 53 (1857), 53–1857.Google Scholar
[13] Lehmer, D. H., Factorization of certain cyclotomic functions. Ann. of Math. 34 (1933), 34–1933.Google Scholar
[14] Louboutin, R., Sur la mesure de Mahler d’un nombre algébrique. C. R. Acad. Sci. Paris 296 (1983), 296–1983.Google Scholar
[15] Mossinghoff, M. J., Polynomials with small Mahler measure.Math. Comp. 67 (1998), 67–1998.Google Scholar
[16] Perron, O., Neue Kriterien für die Irreduzibilität algebraischer Gleichungen. J. Reine Angew. Math. 132 (1907), 132–1907.Google Scholar
[17] Selmer, E. S., On the irreducibility of certain trinomials. Math. Scand. 4 (1956), 4–1956.Google Scholar
[18] Siegel, C. L., Algebraic integers whose conjugates lie in the unit circle. Duke Math. J. 11 (1944), 11–1944.Google Scholar
[19] Smyth, C. J., On the product of conjugates outside the unit circle of an algebraic integer. Bull. London Math. Soc. 3 (1971), 3–1971.Google Scholar
[20] Smyth, C. J., Topics in the theory of numbers. PhD Thesis, University of Cambridge, 1972.Google Scholar
[21] Voutier, P., An effective lower bound for the height of algebraic numbers. Acta Arith. 74 (1996), 74–1996.Google Scholar
[22] Waldschmidt, M., Sur le produit des conjugués extérieurs au cercle unité d’un entier algébrique. Enseign. Math. (2) 26 (1981), 26–1981.Google Scholar
[23] Waldschmidt, M., Diophantine approximation on linear algebraic groups. Grundlehren Math.Wiss. 326, Springer, Berlin-Heidelberg, 2000.Google Scholar