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 Uncanny Subsequence Selections That Generate Normal Numbers

2017 ◽  
Vol 12 (2) ◽  
pp. 65-75 ◽  
Author(s):  
Joseph Vandehey
Keyword(s):  
Real Number ◽  
Arithmetic Progression ◽  
Normal Numbers ◽  
Increasing Sequences

Abstract Given a real number 0.a1a2a3 . . . that is normal to base b, we examine increasing sequences ni so that the number 0.an1an2an3 . . . are normal to base b. Classically, it is known that if the ni form an arithmetic progression, then this will work. We give several more constructions including ni that are recursively defined based on the digits ai. Of particular interest, we show that if a number is normal to base b, then removing all the digits from its expansion which equal (b−1) leaves a base-(b−1) expansion that is normal to base (b − 1)

scholarly journals On Absolutely Normal and Continued Fraction Normal Numbers

2018 ◽  
Vol 2019 (19) ◽  
pp. 6136-6161 ◽  
Author(s):  
Verónica Becher ◽  
Sergio A Yuhjtman
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Main Difficulty ◽  
Normal Numbers ◽  
Continued Fraction Expansion ◽  
Mathematical Operations ◽  
Construction Works

Abstract We give a construction of a real number that is normal to all integer bases and continued fraction normal. The computation of the first n digits of its continued fraction expansion performs in the order of n4 mathematical operations. The construction works by defining successive refinements of appropriate subintervals to achieve, in the limit, simple normality to all integer bases and continued fraction normality. The main difficulty is to control the length of these subintervals. To achieve this we adapt and combine known metric theorems on continued fractions and on expansions in integers bases.

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 Some Normal Numbers Generated by Arithmetic Functions

2015 ◽  
Vol 58 (1) ◽  
pp. 160-173 ◽  
Author(s):  
Paul Pollack ◽  
Joseph Vandehey
Keyword(s):  
Real Number ◽  
Finite Sequence ◽  
Arithmetic Functions ◽  
Normal Numbers ◽  
Euler’S Totient Function ◽  
Sum Of Divisors

AbstractLet g ≥ 2. A real number is said to be g-normal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let φ denote Euler’s totient function, let σ be the sum-of-divisors function, and let λ be Carmichael’s lambda-function. We show that if f is any function formed by composing φ, σ, or λ, then the number0.f(1)f(2)f(3)···obtained by concatenating the base g digits of successive f -values is g-normal. We also prove the same result if the inputs 1,2,3.... are replaced with the primes 2, 3, 5.... The proof is an adaptation of a method introduced by Copeland and Erdõs in 1946 to prove the 10-normality of 0:235711131719...

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.

Coherence and the axioms of confirmation

1955 ◽  
Vol 20 (1) ◽  
pp. 1-28 ◽  
Author(s):  
Abner Shimony
Keyword(s):  
Real Number ◽  
Relative Frequency ◽  
Long Run ◽  
Comparative Concept ◽  
Image Position ◽  
Degree Of Confirmation

It has been pointed out by Carnap that ‘probability’ is an equivocal term, which is used currently in two senses: (i) the degree to which it is rational to believe a hypothesis h on specified evidence e, and (ii) the relative frequency (in an indefinitely long run) of one property of events or things with respect to another. This paper is concerned only with the first of these two senses, which will be referred to as ‘the concept of confirmation,’ in order to avoid equivocation.We may distinguish a quantitative and a comparative concept of confirmation. The general form of statements involving the former isThe degree of confirmation of the proposition h, given the propositione as evidence, is r (where r is a real number between 0 and 1).In this paper the notation for a statement of this form isThe general form of statements involving the comparative concept is The proposition h is equally or less confirmed on e than is the proposition h′ on e′,which may be symbolized asThe following abbreviations will also be used in discussing comparative confirmation:The former of these two abbreviations means intuitivelyThe proposition h is less confirmed on evidence e than h′ is on e′. The latter meansThe proposition h is confirmed on evidence e to the same degree that h′ is on e′.

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

Riesz products, Hausdorff dimension and normal numbers

1987 ◽  
Vol 101 (3) ◽  
pp. 529-540 ◽  
Author(s):  
Gavin Brown ◽  
William Moran ◽  
Charles E. M. Pearce
Keyword(s):  
Hausdorff Dimension ◽  
Real Number ◽  
Normal Numbers ◽  
Riesz Products

We say a real number x is normal to base r if the sequence is uniformly distributed modulo [0, 1]. Pollington[10] has proved the following result concerning the normality of numbers.

Sign in / Sign up

Export Citation Format

Share Document