scholarly journals A curiosity About (−1)[e] +(−1)[2e] + ··· +(−1)[Ne]

2020 ◽  
Vol 15 (2) ◽  
pp. 1-8
Author(s):  
Francesco Amoroso ◽  
Moubinool Omarjee
Keyword(s):  
Real Number ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Partial Quotients

AbstractLet α be an irrational real number; the behaviour of the sum SN (α):= (−1)[α] +(−1)[2α] + ··· +(−1)[Nα] depends on the continued fraction expansion of α/2. Since the continued fraction expansion of \sqrt 2 /2 has bounded partial quotients, {S_N}\left( {\sqrt 2 } \right) = O\left( {\log \left( N \right)} \right) and this bound is best possible. The partial quotients of the continued fraction expansion of e grow slowly and thus {S_N}\left( {2e} \right) = O\left( {{{\log {{\left( N \right)}^2}} \over {\log \,\log {{\left( N \right)}^2}}}} \right), again best possible. The partial quotients of the continued fraction expansion of e/2 behave similarly as those of e. Surprisingly enough 1188.

scholarly journals Renewal-type limit theorem for continued fractions with even partial quotients

2009 ◽  
Vol 29 (5) ◽  
pp. 1451-1478 ◽  
Author(s):  
FRANCESCO CELLAROSI
Keyword(s):  
Limit Theorem ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Type Theorem ◽  
Natural Extension ◽  
Gauss Map ◽  
Continued Fraction Expansion ◽  
Special Flow ◽  
Partial Quotients ◽  
Continued Fraction Expansions

AbstractWe prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by Schweiger [Continued fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg4 (1982), 59–70; On the approximation by continues fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg1–2 (1984), 105–114] and studied also by Kraaikamp and Lopes [The theta group and the continued fraction expansion with even partial quotients. Geom. Dedicata59(3) (1996), 293–333]. Our main result is proven following the strategy used by Sinai and Ulcigrai [Renewal-type limit theorem for the Gauss map and continued fractions. Ergod. Th. & Dynam. Sys.28 (2008), 643–655] in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow.

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.

Slow increasing functions and the largest partial quotients in continued fraction expansions

2016 ◽  
Vol 164 (1) ◽  
pp. 1-14 ◽  
Author(s):  
JINHUA CHANG ◽  
HAIBO CHEN
Keyword(s):  
Level Sets ◽  
Limit Theorems ◽  
Continued Fraction ◽  
Positive Function ◽  
Continued Fraction Expansion ◽  
Hausdorff Dimensions ◽  
Partial Quotients ◽  
Continued Fraction Expansions ◽  
Increasing Functions

AbstractLet 0 ⩽ α ⩽ ∞ and ψ be a positive function defined on (0, ∞). In this paper, we will study the level sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) which are related respectively to the sequence of the largest digits among the first n partial quotients {Ln(x)}n≥1, the increasing sequence of the largest partial quotients {Bn(x)}n⩾1 and the sequence of successive occurrences of the largest partial quotients {Tn(x)}n⩾1 in the continued fraction expansion of x ∈ [0,1) ∩ ℚc. Under suitable assumptions of the function ψ, we will prove that the sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) are all of full Hausdorff dimensions for any 0 ⩽ α ⩽ ∞. These results complement some limit theorems given by J. Galambos [4] and D. Barbolosi and C. Faivre [1].

A DIMENSIONAL RESULT ON THE PRODUCT OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

2021 ◽  
pp. 1-29
Author(s):  
LINGLING HUANG ◽  
CHAO MA
Keyword(s):  
Growth Rate ◽  
Hausdorff Dimension ◽  
Natural Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Continued Fraction Expansion ◽  
Nonnegative Number ◽  
Partial Quotients

Abstract This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set: $$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$ where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$ ) shown by Hussain, Kleinbock, Wadleigh and Wang.

scholarly journals The length of the continued fraction expansion for a class of rational functions in

1991 ◽  
Vol 34 (1) ◽  
pp. 7-17 ◽  
Author(s):  
Arnold Knopfmacher
Keyword(s):  
Finite Field ◽  
Continued Fraction ◽  
Rational Functions ◽  
Euclidean Algorithm ◽  
Continued Fraction Expansion ◽  
Partial Quotients ◽  
Fraction Algorithm

A study is made of the length L(h, k) of the continued fraction algorithm for h/k where h and k are co-prime polynomials in a finite field. In addition we investigate the sum of the degrees of the partial quotients in this expansion for h/k, h, k in . The above continued fraction is determined by means of the Euclidean algorithm for the polynomials h, k in .

scholarly journals Non-periodic continued fractions in hyperelliptic function fields

2001 ◽  
Vol 64 (2) ◽  
pp. 331-343 ◽  
Author(s):  
Alfred J. van der Poorten
Keyword(s):  
Exponential Growth ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Function Fields ◽  
Square Root ◽  
90Th Birthday ◽  
Continued Fraction Expansion ◽  
Generic Polynomial ◽  
Partial Quotients ◽  
Hyperelliptic Function

Dedicated to George Szekeres on his 90th birthdayWe discuss the exponential growth in the height of the coefficients of the partial quotients of the continued fraction expansion of the square root of a generic polynomial.

scholarly journals On the Quantitative Metric Theory of Continued Fractions in Positive Characteristic

2018 ◽  
Vol 61 (1) ◽  
pp. 283-293
Author(s):  
Poj Lertchoosakul ◽  
Radhakrishnan Nair
Keyword(s):  
Finite Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Laurent Series ◽  
Positive Characteristic ◽  
Continued Fraction Expansion ◽  
Formal Laurent Series ◽  
Almost Everywhere ◽  
Partial Quotients ◽  
Image Position

AbstractLet 𝔽q be the finite field of q elements. An analogue of the regular continued fraction expansion for an element α in the field of formal Laurent series over 𝔽q is given uniquely by $$\alpha = A_0(\alpha ) + \displaystyle{1 \over {A_1(\alpha ) + \displaystyle{1 \over {A_2(\alpha ) + \ddots }}}},$$ where $(A_{n}(\alpha))_{n=0}^{\infty}$ is a sequence of polynomials with coefficients in 𝔽q such that deg(An(α)) ⩾ 1 for all n ⩾ 1. In this paper, we provide quantitative versions of metrical results regarding averages of partial quotients. A sample result we prove is that, given any ϵ > 0, we have $$\vert A_1(\alpha ) \ldots A_N(\alpha )\vert ^{1/N} = q^{q/(q - 1)} + o(N^{ - 1/2}(\log N)^{3/2 + {\rm \epsilon }})$$ for almost everywhere α with respect to Haar measure.

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