The Hausdorff dimension of sets of numbers defined by their $Q$-Cantor series expansions

2016 ◽  
Vol 3 (2) ◽  
pp. 163-186 ◽  
Author(s):  
Dylan Airey ◽  
Bill Mance
Keyword(s):  
Hausdorff Dimension ◽  
Series Expansions ◽  
Cantor Series

scholarly journals SHRINKING TARGETS FOR NONAUTONOMOUS DYNAMICAL SYSTEMS CORRESPONDING TO CANTOR SERIES EXPANSIONS

2015 ◽  
Vol 92 (2) ◽  
pp. 205-213 ◽  
Author(s):  
LIOR FISHMAN ◽  
BILL MANCE ◽  
DAVID SIMMONS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Wide Class ◽  
Series Expansions ◽  
Closed Formula ◽  
Nonautonomous Dynamical Systems ◽  
Nonautonomous Dynamical System ◽  
Cantor Series

We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the nonautonomous dynamical system on the interval$[0,1)$, viewed as$\mathbb{R}/\mathbb{Z}$, corresponding to a given method of Cantor series expansion. We also examine a wide class of examples utilising our theorem. In particular, we give a Diophantine approximation interpretation of our scheme.

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 Engel Series Expansions of Laurent Series and Hausdorff Dimensions

2003 ◽  
Vol 75 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Jun Wu
Keyword(s):  
Hausdorff Dimension ◽  
Positive Integer ◽  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Laurent Series ◽  
Series Expansions ◽  
Formal Laurent Series ◽  
Hausdorff Dimensions ◽  
Formal Power

AbstractFor any positive integer q≧2, let Fq be a finite field with q elements, Fq ((z-1)) be the field of all formal Laurent series in an inderminate z, I denote the valuation ideal z-1Fq [[z-1]] in the ring of formal power series Fq ((z-1)) normalized by P(l) = 1. For any x ∈ I, let the series be the Engel expansin of Laurent series of x. Grabner and Knopfmacher have shown that the P-measure of the set A(α) = {x ∞ I: limn→∞ deg an(x)/n = ά} is l when α = q/(q -l), where deg an(x) is the degree of polynomial an(x). In this paper, we prove that for any α ≧ l, A(α) has Hausdorff dimension l. Among other thing we also show that for any integer m, the following set B(m) = {x ∈ l: deg an+1(x) - deg an(x) = m for any n ≧ l} has Hausdorff dimension 1.

scholarly journals Normality of different orders for Cantor series expansions

Nonlinearity ◽  
2017 ◽  
Vol 30 (10) ◽  
pp. 3719-3742
Author(s):  
Dylan Airey ◽  
Bill Mance
Keyword(s):  
Series Expansions ◽  
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.

Sign in / Sign up

Export Citation Format

Share Document