Normal number constructions for Cantor series with slowly growing bases

Dylan Airey; Bill Mance; Joseph Vandehey

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 2, page 465-480
  • ISSN: 0011-4642

Abstract

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Let Q = ( q n ) n = 1 ∞ be a sequence of bases with q i ≥ 2 . In the case when the q i are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose Q -Cantor series expansion is both Q -normal and Q -distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of Q , and from this construction we can provide computable constructions of numbers with atypical normality properties.

How to cite

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Airey, Dylan, Mance, Bill, and Vandehey, Joseph. "Normal number constructions for Cantor series with slowly growing bases." Czechoslovak Mathematical Journal 66.2 (2016): 465-480. <http://eudml.org/doc/280093>.

@article{Airey2016,
abstract = {Let $Q=(q_n)_\{n=1\}^\infty $ be a sequence of bases with $q_i\ge 2$. In the case when the $q_i$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$-Cantor series expansion is both $Q$-normal and $Q$-distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$, and from this construction we can provide computable constructions of numbers with atypical normality properties.},
author = {Airey, Dylan, Mance, Bill, Vandehey, Joseph},
journal = {Czechoslovak Mathematical Journal},
keywords = {Cantor series; normal number; Cantor series; normal numbers; Hausdorff dimension},
language = {eng},
number = {2},
pages = {465-480},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Normal number constructions for Cantor series with slowly growing bases},
url = {http://eudml.org/doc/280093},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Airey, Dylan
AU - Mance, Bill
AU - Vandehey, Joseph
TI - Normal number constructions for Cantor series with slowly growing bases
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 2
SP - 465
EP - 480
AB - Let $Q=(q_n)_{n=1}^\infty $ be a sequence of bases with $q_i\ge 2$. In the case when the $q_i$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$-Cantor series expansion is both $Q$-normal and $Q$-distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$, and from this construction we can provide computable constructions of numbers with atypical normality properties.
LA - eng
KW - Cantor series; normal number; Cantor series; normal numbers; Hausdorff dimension
UR - http://eudml.org/doc/280093
ER -

References

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