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Quantitative transcendence results for numbers associated with Liouville numbers

Author: Deanna M. Caveny
Journal: Proc. Amer. Math. Soc. 120 (1994), 349-357
MSC: Primary 11J82; Secondary 11J85
MathSciNet review: 1165048
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Abstract: In 1937, Franklin and Schneider generalized the Gelfond-Schneider result on the transcendence of $ {\alpha ^\beta }$. They proved the following theorem: If $ \beta $ is an algebraic, irrational number and $ \alpha $ is "suitably well-approximated by algebraic numbers of bounded degree", then $ {\alpha ^\beta }$ is transcendental. In 1964, Feldman established the algebraic independence of $ \alpha $ and $ {\alpha ^\beta }$ under similar conditions. We use results concerning linear forms in logarithms to give quantitative versions of the Franklin-Schneider and Feldman results.

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  • [1] A. Baker, Transcendental number theory, Cambridge Univ. Press, Cambridge and New York, 1975. MR 0422171 (54:10163)
  • [2] W. D. Brownawell, Pairs of polynomials small at a number to certain algebraic powers, Théorie des Nombres: Séminaire Delange-Pisot-Poitou, 17e année: 1975/76, Secrétariat Math., Paris, 1977. MR 0453654 (56:11914)
  • [3] W. D. Brownawell and M. Waldschmidt, The algebraic independence of certain numbers to algebraic powers, Acta. Arith. 32 (1977), 63-71. MR 0429777 (55:2787)
  • [4] P. L. Cijsouw and M. Waldschmidt, Linear forms and simultaneous approximations, Compositio Math. 34 (1977), 173-197. MR 0447130 (56:5445)
  • [5] N. I. Feldman, Arithmetic properties of the solutions of a transcendental equation, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1 (1964), 13-20. MR 0158869 (28:2091)
  • [6] A. O. Gelfond, Transcendental and algebraic numbers, Dover, New York, 1960. MR 0111736 (22:2598)
  • [7] M. Laurent, Indépendance algébrique de nombres de Liouville à des puissances algébriques, Ph.D. thesis, L'Universite Pierre et Marie Curie, Paris, 1977.
  • [8] P. Philippon and M. Waldschmidt, Lower bounds for linear forms in logarithms, New Advances in Transcendence Theory (A. Baker, ed.), Cambridge Univ. Press, New York, 1988, pp. 280-312. MR 972007 (90b:11072)
  • [9] R. Tubbs, A note on some elementary measures of algebraic independence, Proc. Amer. Math. Soc. 109 (1990), 297-304. MR 1007514 (90i:11077)
  • [10] K. Väänänen, On the arithmetic properties of certain values of the exponential function, Stud. Sci. Math. Hungar. 11 (1976), 399-405. MR 554587 (83a:10063)
  • [11] M. Waldschmidt, Simultaneous approximation of numbers connected with the exponential function, J. Austral. Math. Soc. Ser. A 25 (1978), 466-478. MR 0506049 (58:21955)
  • [12] -, Nouvelles méthodes pour minorer des combinaisons linéaires de logarithmes de nombres algébriques, Sém. Théorie Nombres, Bordeaux 3 (1991), 129-185. MR 1116105 (92e:11077)
  • [13] M. Waldschmidt and Y. Zhu, Algebraic independence of certain numbers related to Liouville numbers, Sci. China Ser. A 33 (1990), 257-268. MR 1058178 (91e:11084)

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Keywords: Quantitative, transcendence, Liouville numbers
Article copyright: © Copyright 1994 American Mathematical Society

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