Questions tagged [continued-fractions]
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204
questions
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continued fraction for logarithmic integral
Does the logarithmic integral function $\operatorname{li}(x)$ have the continued fraction expansion
$$\operatorname{li}(x) = \cfrac{x}{\log x -1 -{}} \ \cfrac{1}{\log x -3 -{}} \ \cfrac{4}{\log x -...
1
vote
1
answer
152
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Bounded, aperiodic irrationals with bounded, aperiodic sum
If $q = [q_0;q_1 \dots]$, say $q_i$ is the $i$-th partial quotient of $q$. My question is the following:
Can one construct an explicit example of irrational $r,s > 0$ such that
$\{ 1,r,s\}$ is $\...
1
vote
0
answers
178
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Theoretical invariant distribution of discrete dynamical systems, including the Riemann Zeta map
Update on 3/10/2021: I added Example 5 in the Appendix. This generic example encompasses the Riemann Zeta dynamical system. A simple version of this post, targeted to engineers, machine learning ...
2
votes
1
answer
514
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About generalized continued fractions
Let us consider the sequences $(x_n), (a_n)$, starting with $n=0$ and $x_0\in ]0,1[$, defined by the following generalized Gaussian map:
$$x_{n+1}=\frac{\lambda_n}{x_n^{\alpha_n}}-\Big\lfloor \frac{\...
5
votes
1
answer
326
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Irrationality of $e^{x/y}$
How to prove the following continued fraction of $e^{x/y}$
$${\displaystyle e^{x/y}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+{\cfrac {x^{2}}{18y+\ddots }}}}}}}}}}...
6
votes
1
answer
638
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Algebraic and rational parts of a real number
Let $\alpha$ be a positive real number. Does it make sense to define the closest rational to $\alpha$ as the number $R(\alpha)=\frac{p_1}{p_2}$ such that $p_1,p_2$ are positive co-prime integers ...
8
votes
1
answer
714
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An alternative to continued fraction and applications
This post is inspired by the Numberphile video 2.920050977316, advertising the paper A Prime-Representing Constant by Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, ...
1
vote
0
answers
75
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continued fractions and cusp non-excursions
Consider the modular surface $X:=\mathbb{H^2}/PSL_2(\mathbb{Z})$.
Fix a width-of-cusp parameter $w, 0<w<<1$.
Let $B_w$ be the cusp neighborhood of width $w$. (So $w=1$ corresponds to the ...
17
votes
0
answers
702
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Picture of Lambert's proof that $\pi$ is irrational?
With a suitably generous notion of "picture proof" or "proof without words" or "geometric proof," there do exist such proofs of the irrationality of square roots and even ...
3
votes
0
answers
228
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Asymptotic expansions for the continued fraction $[1,x,x^2,x^3,\cdots]$
The $n$-th convergent is defined as
$$R_n(x) = \frac{P_n(x)}{Q_n(x)}=[1;x,x^2,\cdots,x^n]=1+\frac{1}{x+}\frac{1}{x^2+}\frac{1}{x^3+\cdots}\frac{1}{x^n}$$
where $P_n(x), Q(x)$ are polynomials ...
2
votes
1
answer
114
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Continued fractions, Chebyshev and non-homogenous approximation
In Khinchin's book, "Continued Fractions," he considers the question, given an irrational, $\alpha$, and a real number, $\beta$, how to find integral $x$ and $y$ such that
$$\alpha x - y \...
6
votes
0
answers
233
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Continued fractions and class groups
Let $d$ be a positive integer. It is well-known (due to Lagrange) that the continued fraction of $\sqrt{d}$ is eventually periodic. Moreover, it is known that the equation
$$\displaystyle x^2 - dy^2 = ...
4
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0
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252
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Why is Haven's discovery important?
Today my attention was caught by one of those little stories that appear when you open a certain browser: an inmate achieved a number theoretic breakthrough
It is about continued fractions and I would ...
2
votes
1
answer
155
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On a transcendental number defined as a variation involving the Lambert $W$ function in the nested square root representation of the golden ratio
Define the real number $\xi$ satisfying
$$\xi=\sqrt{1+W\left(1+\sqrt{1+W(1+\ldots)}\right)}\tag{1}$$
where $W(x)$ denotes the main branch of the Lambert $W$ function, as reference I add that Wikipedia ...
1
vote
0
answers
201
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What is the nearest Ford circle for any point in $\mathbb R^2$
I want to draw Ford circles within a "distance Estimated system" (ray marching). Therefore, given a point $(x,y)$ from $\mathbb R^2$, I need the shortest distance to any circle with center $(p/q,1/2q^...
3
votes
1
answer
332
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Proof of continued fraction identity of subfactorial
This question is part of a wider conjecture I have formed with someone which has its roots in Raayoni et al. (2019). The subfactorial function can be written as $$!n=\frac{n!}e+\frac{(-1)^n}{n+2-\...
7
votes
0
answers
273
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Possible Birkhoff spectra for irrational rotations
Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the unit circle (think of it as of the interval $[0,1)$ with endpoints identified). Assume that $\alpha$ is irrational and consider the rotation by $\alpha$, ...
3
votes
1
answer
267
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Distribution of partial quotients of $\sqrt{d}$ with $X < d \leq 2X$
Recall that for a real number $\alpha$, it has a continued fraction expansion usually written as
$$\displaystyle \alpha = [a_0; a_1, a_2, \cdots].$$
Moreover, $\alpha$ is rational if and only if its ...
1
vote
0
answers
61
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Maximal orders in Clifford algebras
Let
$$
\mathcal{C}_n(R)=R\langle e_1,\ldots,e_n\rangle/(\{e_i^2+1\}, \{e_ie_j+e_je_i:i\neq j\})
$$
be the Clifford algebra for the negative definite quadratic form $-\sum_ix_i^2$ obtained by adjoining ...
6
votes
1
answer
506
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Are the coefficients of certain product of Rogers-Ramanujan Continued Fraction non-negative?
Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$
The following equality is famous:
$$\cfrac{q^{1/5}}{R(q)} = \prod_{k>0} \cfrac{(1-q^{5k-2})(1-q^{...
7
votes
2
answers
513
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Average number of iterations for the Euclidean algorithm to terminate
Let $N$ be a positive integer and $0 \leq s < N$.
We try to divide $s$ into $N$ using the Euclidean algorithm:
$N = q_1 s + r_1 $
$r = q_2 r_1 + r_2 $
$\vdots$
$r_{K-1} = q_{K-1} r_K$
If we ...
7
votes
2
answers
191
views
Evaluation of hypergeometric type continued fraction
Is there a (possibly hypergeometric-type) explicit evaluation of the
continued fraction
$$a-\dfrac{1.(c+d)}{2a-\dfrac{2.(2c+d)}{3a-\dfrac{3.(3c+d)}{4a-\ddots}}}$$
Even the special case $d=0$, $a=1$ ...
5
votes
2
answers
744
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Does using continued fractions work to give a homeomorphism $\mathbb{Q}^+ \rightarrow (\mathbb{Q}^+)^2$?
Let $\mathbb{Q}$ be the topological space of rational numbers (with topology induced by inclusion in the real line) and let $\mathbb{Q}^+$ be the set of positive ($x>0$) rationals.
I'm looking for ...
3
votes
0
answers
152
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Why does division parallelize but not continued fractions and is there an analog of multiplication to continued fractions?
All the basic arithmetic operations $\times,+,/,-$ can be parallelized. However continued fraction representation of a rational number is not parallelized. The process of Euclid's algorithm looks ...
9
votes
3
answers
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Have new conjectures generated by the Ramanujan machine been proven?
Recently the following preprint was published with new automatically generated conjectures on generalizes continuous fractions, e.g., for the Euler constant: Raayoni, Pisha, Manor, Mendlovic, Haviv, ...
18
votes
1
answer
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Divergent Series & Continued Fraction (from Gauss' Mathematical Diary)
I've asked that question before on History of Science and Mathematics but haven't received an answer
Does someone have a reference or further explanation on Gauß' entry from May 24, 1796 in his ...
8
votes
2
answers
398
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Riemann-Hilbert and orthogonal polynomials
Sorry for perhaps naive questions, I am not at all a specialist in the subject
but I need it for my research.
I know that there are close relations between Riemann-Hilbert problems and
orthogonal ...
34
votes
2
answers
1k
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Representations of $\zeta(3)$ as continued fractions involving cubic polynomials
$\zeta(3)$ has at least two well-known representations of the form $$\zeta(3)=\cfrac{k}{p(1) - \cfrac{1^6}{p(2)- \cfrac{2^6}{ p(3)- \cfrac{3^6}{p(4)-\ddots } }}},$$
where $k\in\mathbb Q$ and $p$ is a ...
5
votes
0
answers
107
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"middle" partial denominator in continued fraction expansion of square roots
Suppose $d$ is a positive integer that is not a perfect square such that the negative Pell equation, $x^{2}-dy^{2}=-1$ has no solution. Then we know the minimal period of the continued fraction ...
8
votes
1
answer
175
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Limit of quotients of elements of special Fibonacci matrices
Let $F_n$ be the $n$-th Fibonacci number, started with $F_0=0,F_1=1$, and consider the matrices
$$M_n=\pmatrix{F_{n+3} & F_{n+1} \\ F_{n+2} & F_{n}}.$$
Let
$$\pmatrix{\alpha_n & \beta_n \\...
1
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0
answers
90
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Bound for truncation error of continued fraction for $E_1(z)$
Let $z \in \mathbb C \setminus(-\infty,0)$. It is known that
$$E_1(z) = \cfrac{e^{-z}}{z+\cfrac{1}{1+\cfrac{1}{z+\cfrac{2}{1+\cfrac{2}{z+\cfrac{3}{1+\cdots}}}}}}.$$
For example, see http://functions....
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vote
1
answer
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Distinctness of quadratic surd continued fraction convergent ratio limit
In this question on math.stackexchange.com I have made two conjectures the first of which I have proved. The second has not been settled. I post it here to seek a proof.
Given a quadratic surd $\sqrt ...
0
votes
0
answers
63
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Reduction of a Jacobi-type continued fraction
I am trying to reduce the following Jacobi-like continued fraction(or J-fraction):
$$f(z)=z+K_{n=1}^{\infty}\frac{R_{n}k^2}{z+Q_{n}}$$
where,
$$R_{n}=n\left(n-\frac{1}{2}\right),\; Q_n=n\left(n+m+\...
6
votes
0
answers
503
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Theory of Irrational Tangles?
According to one possible definition, an $n$-tangle $T$ is a subset $T \subseteq \Bbb{R}^2\times [0,1] =: X$ that is homeomorphic to a disjoint union $[0,1] \times n := [0,1] \amalg \ldots \amalg [0,1]...
6
votes
2
answers
398
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Periods of the continued fraction expansions of Galois-conjugate quadratic-irrationals
Question: Given a quadratic irrational $x = a + b\sqrt{D}$ ($a,b \in \Bbb{Q}$, $D \in \Bbb{N}_{> 0}$ square-free) and its Galois conjugate $x' = a - b\sqrt{D}$, is it true that the continued ...
5
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0
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Transcendental Continued Fractions
Liouville famously showed the existence of a transcendental number by considering $\alpha =\sum\limits_{n=0}^\infty10^{-n!}$ and showing that it did not satisfy 'Liouville's criterion' $|\alpha-p/q|\...
0
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0
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Different characterizations of Liouville numbers
Usually, Liouville numbers are defined as follows:
$x$ is Liouville if for ever $i\in\mathbb N$ there exist $n,m\in\mathbb Z$ such that
\begin{equation}
\left|x-\frac nm\right|<\frac1{m^i}.
\end{...
3
votes
1
answer
357
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Matrix continued fractions
I am aware of the classical continued fraction in the field of real numbers, but recently I have come across the term matrix continued fraction and when I checked on the internet there are varieties ...
1
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2
answers
243
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Does the set of Diophantine $m$-tuples has full measure?
We say that an $m$-tuple $\omega=(\omega_1,\ldots,\omega_m)$ satisfies the Diophantine condition of order $\nu \geq 0$ if there is a constant $C>0$ such that for all natural $q$ and integer $p_1,\...
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Relation between a continued fraction and partitions
I am interested in the continued fraction
$$\sum\limits_k {{z^{{2^k} - 1}}} = \frac{1}{{1 - \frac{{{T_0}z}}{{1 - \frac{{{T_1}z}}{{1 - \frac{{{T_2}z}}{{1 -{ \ddots }}}}}}}}}.$$
OEIS A104977 states ...
1
vote
1
answer
185
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Simultaneous Diophantine Condition and Growth Rate of Convergents Denominators
Let $\omega=(\omega_1,\ldots,\omega_{m})$ be an $m$-tuple of real numbers. Let $|\omega|_{m}:=\sup\limits_{1 \leq j \leq m}|\omega_j|_{1}$ be a metric on flat torus $\mathbb{T}^{m}=\mathbb{R}^{m}/\...
0
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0
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whether the quotient of continued fraction of algebraic irrational number is bounded or not is similar or equivalent to Collatz conjecture?
I vaguely recall that whether the quotient of continued fraction of algebraic irrational number is bounded or not is similar or equivalent to Collatz conjecture, could any one give the reference? or ...
10
votes
1
answer
226
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Distribution of good diophantine approximations
Let $\langle x \rangle: \mathbb{R} \to (-1/2,1/2]$ be the periodic function with period $1$ which is $x$ for $x \in (-1/2,1/2]$. Is there some function $D(a,b)$ of real numbers $a<b$ such that, for ...
3
votes
1
answer
390
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Continued Fraction of Random Variables
So this is my first post in mathoverflow. I posted this problem in Mathstack, an I've also
put a bounty on it, but did not get any response. If anyone can at least point out a reference on this ...
6
votes
2
answers
2k
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Distribution of distances of successive zeros of $f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$
Let $\alpha$ and $\beta$ be incommensurate real numbers.
Consider the function
$f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$
and its positive zeros $x_k(\alpha,\beta)$.
Fix $\alpha$ and ...
14
votes
0
answers
344
views
Quasiperiodic continued fractions
Is anything known about continued fractions in which the sequence of integers is quasiperiodic?
Quasiperiodic is meant here in the sense of 1D quasicrystals. For example, draw an irrationally-sloped ...
6
votes
3
answers
588
views
Irrationality of generalized continued fractions
An infinite simple continued fraction
$$\frac{1}{b_1 + \frac{1}{b_2 + \frac{1}{b_3+\dots}}} (b_i\in\mathbb Z)$$
is irrational. Now for a generalized continued fraction:
$$\frac{a_1}{b_1 + \frac{a_2}...
2
votes
0
answers
150
views
Has anybody studied continued fractions in function spaces?
For the text below, define $f^\infty(x) = \lim_{n\to\infty} f^n(x)$ where $f^n = \underbrace{f \circ \ldots \circ f}_{n}$.
Usually 'continued fraction' means continued fraction in $\mathbb{R}$. For ...
9
votes
1
answer
353
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Some nice functional equations for $q$-continued fractions
Given $\large q=e^{2\pi i \tau}$. Define,
$$\alpha(\tau) = \sqrt2\,q^{1/8}\prod_{n=1}^\infty\frac{ (1-q^{4n-1})(1-q^{4n-3})}{(1-q^{4n-2})(1-q^{4n-2})}$$
$$\beta(\tau) = q^{1/5}\prod_{n=1}^\infty\frac{ ...
7
votes
0
answers
166
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The Heine $q$-continued fraction
Let $q=e^{2\pi i \tau}$. The Heine continued fraction is $$H_2(\tau)=\frac1{q^{1/24}}\frac{\eta(2\tau)}{\eta(\tau)} =1+\cfrac{q}{1-q+\cfrac{q^3-q^2}{1+\cfrac{q^5-q^3}{1+\cfrac{q^7-q^4}{1+\ddots}}}}$$
...