The irrationality exponents of computable numbers
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- by Verónica Becher, Yann Bugeaud and Theodore A. Slaman PDF
- Proc. Amer. Math. Soc. 144 (2016), 1509-1521 Request permission
Abstract:
We establish that there exist computable real numbers whose irrationality exponent is not computable.References
- A. S. Besicovitch, Sets of Fractional Dimensions (IV): On Rational Approximation to Real Numbers, J. London Math. Soc. 9 (1934), no. 2, 126–131. MR 1574327, DOI 10.1112/jlms/s1-9.2.126
- Yann Bugeaud, Diophantine approximation and Cantor sets, Math. Ann. 341 (2008), no. 3, 677–684. MR 2399165, DOI 10.1007/s00208-008-0209-4
- Kenneth Falconer, Fractal geometry, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. Mathematical foundations and applications. MR 2118797, DOI 10.1002/0470013850
- V. Jarník, Zur metrischen theorie der diophantischen approximation, Prace Mat.-Fiz. 36 (1928/1929), 91–106.
- Vojtěch Jarník, Über die simultanen diophantischen Approximationen, Math. Z. 33 (1931), no. 1, 505–543 (German). MR 1545226, DOI 10.1007/BF01174368
- Wolfgang M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR 568710
- J. O. Shallit, Simple continued fractions for some irrational numbers. II, J. Number Theory 14 (1982), no. 2, 228–231. MR 655726, DOI 10.1016/0022-314X(82)90047-6
- Robert I. Soare, Recursive theory and Dedekind cuts, Trans. Amer. Math. Soc. 140 (1969), 271–294. MR 242667, DOI 10.1090/S0002-9947-1969-0242667-4
- Robert I. Soare, Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987. A study of computable functions and computably generated sets. MR 882921, DOI 10.1007/978-3-662-02460-7
Additional Information
- Verónica Becher
- Affiliation: Departamento de Computacion, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, C1428EGA Ciudad Autónoma de Buenos Aires, Argentina
- MR Author ID: 368040
- Email: vbecher@dc.uba.ar
- Yann Bugeaud
- Affiliation: UFR de Mathématique et d’Informatique, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France
- Email: bugeaud@math.unistra.fr
- Theodore A. Slaman
- Affiliation: Department of Mathematics, 970 Evans Hall, University of California Berkeley, Berkeley, California 94720
- MR Author ID: 163530
- Email: slaman@math.berkeley.edu
- Received by editor(s): August 22, 2014
- Received by editor(s) in revised form: May 13, 2015
- Published electronically: October 6, 2015
- Communicated by: Matthew A. Papanikolas
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 1509-1521
- MSC (2010): Primary 11J04; Secondary 03Dxx
- DOI: https://doi.org/10.1090/proc/12841
- MathSciNet review: 3451228