Artículo
Abstract:
We demonstrate the full logical independence of normality to multiplicatively independent bases. This establishes that the set of bases to which a real number can be normal is not tied to any arithmetical properties other than multiplicative dependence. It also establishes that the set of real numbers which are normal to at least one base is properly at the fourth level of the Borel hierarchy, which was conjectured by A. Ditzen 20 years ago.We further show that the discrepancy functions for multiplicatively independent bases are pairwise independent. In addition, for any given set of bases closed under multiplicative dependence, there are real numbers that are normal to each base in the given set, but not simply normal to any base in its complement. This answers a question first raised by Brown, Moran and Pearce. ©2014 London Mathematical Society.
Registro:
| Documento: | Artículo |
| Título: | On the normality of numbers to different bases |
| Autor: | Becher, V.; Slaman, T.A. |
| Filiación: | Departamento de Computación, Facultad de Ciencias Exactas Y Naturales, Universidad de Buenos Aires and CONICET, Pabellon I, Ciudad Universitaria, Buenos Aires, 1428, Argentina Department of Mathematics, University of California, Berkeley 719 Evans Hall #3840, Berkeley, CA 94720-3840, United States |
| Año: | 2014 |
| Volumen: | 90 |
| Número: | 2 |
| Página de inicio: | 472 |
| Página de fin: | 494 |
| DOI: | http://dx.doi.org/10.1112/jlms/jdu035 |
| Título revista: | Journal of the London Mathematical Society |
| Título revista abreviado: | J. Lond. Math. Soc. |
| ISSN: | 00246107 |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00246107_v90_n2_p472_Becher |
Referencias:
- Brown, G., Moran, W., Pearce, C.E.M., 'Riesz products and normal numbers' (1985) J. London Math. Soc., 32 (2), pp. 12-18
- Bugeaud, Y., Distribution modulo one and diophantine approximation (2012) Cambridge Tracts in Mathematics, 193. , Cambridge University Press, Cambridge, UK
- Ki, H., Linton, T., 'Normal numbers and subsets of N with given densities' (1994) Fund. Math., 144, pp. 163-179
- Kuipers, L., Niederreiter, H., (2006) Uniform Distribution of Sequences, , Dover Publications, Inc. New York
- Moran, W., Pearce, C.E.M., 'Discrepancy results for normal numbers', Miniconferences on harmonic analysis and operator algebras (1988) Proceedings of the Centre for Mathematics and its Applications, 16, pp. 203-210. , Australian National University, Canberra
- Pollington, A.D., 'The Hausdorff dimension of a set of normal numbers' (1981) Pacific J. Math., 95, pp. 193-204
- Rogers, H., Jr., (1987) Theory of Recursive Functions and Effective Computability, , 2nd edn (MIT Press, Cambridge, MA
- Schmidt, W.M., 'On normal numbers' (1960) Pacific J. Math., 10, pp. 661-672
- Schmidt, W.M., 'Uber die Normalitat von Zahlen zu verschiedenen Basen' (1961) Acta Arith., 7, pp. 299-309
Citas:
---------- APA ----------
Becher, V. & Slaman, T.A. (2014) . On the normality of numbers to different bases. Journal of the London Mathematical Society, 90(2), 472-494.http://dx.doi.org/10.1112/jlms/jdu035
---------- CHICAGO ----------
Becher, V., Slaman, T.A. "On the normality of numbers to different bases" . Journal of the London Mathematical Society 90, no. 2 (2014) : 472-494.http://dx.doi.org/10.1112/jlms/jdu035
---------- MLA ----------
Becher, V., Slaman, T.A. "On the normality of numbers to different bases" . Journal of the London Mathematical Society, vol. 90, no. 2, 2014, pp. 472-494.http://dx.doi.org/10.1112/jlms/jdu035
---------- VANCOUVER ----------
Becher, V., Slaman, T.A. On the normality of numbers to different bases. J. Lond. Math. Soc. 2014;90(2):472-494.http://dx.doi.org/10.1112/jlms/jdu035
