On the normality of numbers to different bases

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 numbe...

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Detalles Bibliográficos
Autores principales: Becher, V., Slaman, T.A.
Formato: JOUR
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00246107_v90_n2_p472_Becher
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_00246107_v90_n2_p472_Becher2023-10-03T14:35:16Z On the normality of numbers to different bases Becher, V. Slaman, T.A. 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. Fil:Becher, V. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00246107_v90_n2_p472_Becher
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
description 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.
format JOUR
author Becher, V.
Slaman, T.A.
spellingShingle Becher, V.
Slaman, T.A.
On the normality of numbers to different bases
author_facet Becher, V.
Slaman, T.A.
author_sort Becher, V.
title On the normality of numbers to different bases
title_short On the normality of numbers to different bases
title_full On the normality of numbers to different bases
title_fullStr On the normality of numbers to different bases
title_full_unstemmed On the normality of numbers to different bases
title_sort on the normality of numbers to different bases
url http://hdl.handle.net/20.500.12110/paper_00246107_v90_n2_p472_Becher
work_keys_str_mv AT becherv onthenormalityofnumberstodifferentbases
AT slamanta onthenormalityofnumberstodifferentbases
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