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dc.contributor.author
Christoph Aistleitne
dc.contributor.author
Becher, Veronica Andrea
dc.contributor.author
Olivier Carton
dc.date.available
2020-12-23T17:32:22Z
dc.date.issued
2019-04
dc.identifier.citation
Christoph Aistleitne; Becher, Veronica Andrea; Olivier Carton; Normal numbers with digit dependencies; American Mathematical Society; Transactions Of The American Mathematical Society; 113; 2; 4-2019; 169-178
dc.identifier.issn
0002-9947
dc.identifier.uri
http://hdl.handle.net/11336/121152
dc.description.abstract
We give metric theorems for the property of Borel normality for real numbers under the assumption of digit dependencies in their expansion in a given integer base. We quantify precisely how much digit dependence can be allowed such that, still, almost all real numbers are normal. Our theorem states that almost all real numbers are normal when at least slightly more than $log log n$ consecutive digits with indices starting at position $n$ are independent. As the main application, we consider the Toeplitz set $T_P$, which is the set of all sequences $a_1a_2 ldots $ of symbols from ${0, ldots, b-1}$ such that $a_n$ is equal to $a_{pn}$, for every $p$ in $P$ and $n=1,2,ldots$. Here~$b$ is an integer base and~$P$ is a finite set of prime numbers. We show that almost every real number whose base $b$ expansion is in~$T_P$ is normal to base~$b$. In the case when $P$ is the singleton set ${2}$ we prove that more is true: almost every real number whose base $b$ expansion is in $T_P$ is normal to all integer bases. We also consider the Toeplitz transform which maps the set of all sequences to the set $T_P$ and we characterize the normal sequences whose Toeplitz transform is normal as well.
dc.format
application/pdf
dc.language.iso
eng
dc.publisher
American Mathematical Society
dc.rights
info:eu-repo/semantics/restrictedAccess
dc.rights.uri
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.subject
DISTRIBUTION MODULO ONE
dc.subject
NORMAL NUMBERS
dc.subject.classification
Ciencias de la Computación
dc.subject.classification
Ciencias de la Computación e Información
dc.subject.classification
CIENCIAS NATURALES Y EXACTAS
dc.title
Normal numbers with digit dependencies
dc.type
info:eu-repo/semantics/article
dc.type
info:ar-repo/semantics/artículo
dc.type
info:eu-repo/semantics/publishedVersion
dc.date.updated
2020-11-18T17:32:14Z
dc.journal.volume
113
dc.journal.number
2
dc.journal.pagination
169-178
dc.journal.pais
Estados Unidos
dc.journal.ciudad
Providence
dc.description.fil
Fil: Christoph Aistleitne. Technical University Graz; Austria
dc.description.fil
Fil: Becher, Veronica Andrea. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentina
dc.description.fil
Fil: Olivier Carton. Université Paris Diderot; Francia
dc.journal.title
Transactions Of The American Mathematical Society
dc.relation.alternativeid
info:eu-repo/semantics/altIdentifier/doi/http://dx.doi.org/10.1090/tran/7706
dc.relation.alternativeid
info:eu-repo/semantics/altIdentifier/url/https://www.ams.org/journals/tran/2019-372-06/S0002-9947-2018-07706-6/
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