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T-numbers form an M0 set

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  26 February 2010

W. Moran ,
C. E. M. Pearce  and
A. D. Pollington
W. Moran
Affiliation:
School of Information Science and Technology, Flinders University of South Australia, GPO Box 2100, Adelaide SA 5001, Australia.
C. E. M. Pearce
Affiliation:
Department of Applied Mathematics, The University of Adelaide, GPO Box 298, Adelaide SA 5001, Australia.
A. D. Pollington
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA.
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Extract

Abstract. We show that the set of T-numbers in Mahler's classification of transcendental numbers supports a measure whose Fourier transform vanishes at infinity. A similar argument shows that U-numbers also support such a measure.

MSC classification

Secondary: 11J81: Transcendence (general theory)
Type
Research Article
Information
Mathematika , Volume 39 , Issue 1 , June 1992 , pp. 18 - 24
Copyright
Copyright © University College London 1992

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References

1.Baker, A.. Transcendental Number Theory (Cambridge University Press, 1975).CrossRefGoogle Scholar
2.Baker, R.. On approximation with algebraic numbers of bounded degree. Mathematika, 23 (1976), 18–31.CrossRefGoogle Scholar
3.Ivashev-Musatov, O. S.. The coefficients of trigonometric null series (in Russian). Izv. Akad. Nauk SSSR, Soviet Math., 2 (1957), 559–578.Google Scholar
4.Kahane, J.-P. and Katznelson, Y.. Sur les ensembles d'unicité U(ε) de Zygmund. C.R. Acad. Sci. Paris, 227 (1973), 893–895.Google Scholar
5.Kasch, F. and Volkmann, B.. Zur Mahlerschen Vermutung über S-Zahlen. Math. Annalen, 136 (1958), 442–453.CrossRefGoogle Scholar
6.Kaufman, R.. Continued fractions and Fourier transforms. Mathematika, 27 (1980), 220–229.CrossRefGoogle Scholar
7.Kaufman, R.. On the theorem of Jarnik and Besicovitch. Acta Mathematika, 39 (1981), 265–267.Google Scholar
8.Koksma, J.. Über die Mahlersche Klasseneinteilung der transendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen. Monatsh. Math. Phys., 48 (1939), 176–189.CrossRefGoogle Scholar
9.Korner, T. W.. Some results on Kronecker, Dirichlet and Helson sets. Ann. Inst. Fourier, 20 (1970), 219–324.CrossRefGoogle Scholar
10.LeVeque, W. J.. On Mahler's U-numbers. J. London Math. Soc., 28 (1953), 220–229.CrossRefGoogle Scholar
11.Mahler, K.. Zur Approximation der Exponentialfunktion und des Logarithmus I. Journal Reine Angew. Math., 166 (1932), 118–136.CrossRefGoogle Scholar
12.Mahler, K.. Uber das Mass der Menge aller S-Zahlen. Math. Ann., 106 (1932), 131–139.CrossRefGoogle Scholar
13.Schmidt, W. M.. T-numbers do exist. Symposia Math. IV, Inst. Naz. di Aha Math., Rome 1968 (Academic Press 1970), 3–26.Google Scholar
14.Schmidt, W. M.. Mahler's T-numbers. Proc. of Symposia in Pure Math., 20 (Institute on Number Theory, Amer. Math. Soc, 1969), 275–286.Google Scholar
15.Schmidt, W. M.. Approximation to algebraic numbers (Monographic No 19 de L'Enseignement Mathématique, Université, Genève, 1972).Google Scholar