scholarly journals Construction of normal numbers with respect to the Q-Cantor series expansion for certain Q

2011 ◽  
Vol 148 (2) ◽  
pp. 135-152 ◽  
Author(s):  
Bill Mance
Keyword(s):  
Series Expansion ◽  
Normal Numbers ◽  
Cantor Series

scholarly journals BETA CANTOR SERIES EXPANSION AND ADMISSIBLE SEQUENCES

2020 ◽  
Vol 60 (3) ◽  
pp. 214-224
Author(s):  
Jonathan Caalim ◽  
Shiela Demegillo
Keyword(s):  
Real Number ◽  
Series Expansion ◽  
Numeration System ◽  
Integer Sequence ◽  
Cantor Series ◽  
Admissible Sequences

We introduce a numeration system, called the <em>beta Cantor series expansion</em>, that generalizes the classical positive and negative beta expansions by allowing non-integer bases in the Q-Cantor series expansion. In particular, we show that for a fix $\gamma \in \mathbb{R}$ and a sequence $B=\{\beta_i\}$ of real number bases, every element of the interval $x \in [\gamma,\gamma+1)$ has a <em>beta Cantor series expansion</em> with respect to B where the digits are integers in some alphabet $\mathcal{A}(B)$. We give a criterion in determining whether an integer sequence is admissible when $B$ satisfies some condition. We provide a description of the reference strings, namely the expansion of $\gamma$ and $\gamma+1$, used in the admissibility criterion.

scholarly journals On the Hausdorff dimension faithfulness and the Cantor series expansion

2020 ◽  
Vol 26 (4) ◽  
pp. 298-310
Author(s):  
S. Albeverio ◽  
Ganna Ivanenko ◽  
Mykola Lebid ◽  
Grygoriy Torbin
Keyword(s):  
Hausdorff Dimension ◽  
Series Expansion ◽  
Cantor Series

scholarly journals Cantor series constructions of sets of normal numbers

2012 ◽  
Vol 156 (3) ◽  
pp. 223-245 ◽  
Author(s):  
Bill Mance
Keyword(s):  
Normal Numbers ◽  
Cantor Series

scholarly journals Number theoretic applications of a class of Cantor series fractal functions, II

2015 ◽  
Vol 11 (02) ◽  
pp. 407-435 ◽  
Author(s):  
Brian Li ◽  
Bill Mance
Keyword(s):  
Theoretical Result ◽  
Real Number ◽  
Arithmetic Progression ◽  
Relative Frequency ◽  
Series Expansions ◽  
Basic Fact ◽  
Normal Numbers ◽  
Fractal Functions ◽  
Cantor Series

It is well known that all numbers that are normal of order k in base b are also normal of all orders less than k. Another basic fact is that every real number is normal in base b if and only if it is simply normal in base bkfor all k. This may be interpreted to mean that a number is normal in base b if and only if all blocks of digits occur with the desired relative frequency along every infinite arithmetic progression. We reinterpret these theorems for the Q-Cantor series expansions and show that they are no longer true in a particularly strong way. The main theoretical result of this paper will be to reduce the problem of constructing normal numbers with certain pathological properties to the problem of solving a system of Diophantine relations.

scholarly journals Normal Numbers and Limit Computable Cantor Series

2017 ◽  
Vol 58 (2) ◽  
pp. 215-220 ◽  
Author(s):  
Achilles Beros ◽  
Konstantinos Beros
Keyword(s):  
Normal Numbers ◽  
Cantor Series
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