All Questions
25 questions
8
votes
2
answers
178
views
Distribution of traces and max entries of words of fixed length in $\operatorname{SL}_2(\mathbb{N})$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\tr{\mathsf{tr}}$$\SL_2(\mathbb{N})$ is a free monoid on the generators
$$
L=\begin{pmatrix}1&0\\1&1\end{pmatrix},\quad R=\begin{pmatrix}1&1\...
- 609
0
votes
1
answer
292
views
Factorization trees and (continued) fractions?
This question is inspired by trying to understand the lexicographic sorting of the natural numbers in the fractal at this question:
Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , ...
- 3,064
2
votes
1
answer
148
views
A bijection between odd natural integers and compositions
Given an odd natural integer $2a-1$ with $a\geq 1$, associate to it recursively the composition $\psi(1)=\emptyset$ and $\psi(2^{-n}a)+(n+\delta_{>1}(m))$ if $a=2^n m$ with $m$ odd where $\delta_{&...
- 17.9k
3
votes
1
answer
206
views
The growth of certain continued fractions
I was recently looking into an old problem of Hardy which studies the distribution of integers of the form $2^a 3^b \leq x$, where $a,b\geq 0$. Letting $N(x)$ denote the number of pairs $(a,b)$ ...
- 1,990
3
votes
1
answer
270
views
Distance formula for continued fractions
In the book Neverending fractions from Borwein, van der Poorten, Shallit and Zudilin, there is the so called distance formula (Theorem 2.45, p. 43) stated:
$$\alpha_1\alpha_2\cdot...\cdot\alpha_n=\...
- 406
2
votes
1
answer
162
views
On a transcendental number defined as a variation involving the Lambert $W$ function in the nested square root representation of the golden ratio
Define the real number $\xi$ satisfying
$$\xi=\sqrt{1+W\left(1+\sqrt{1+W(1+\ldots)}\right)}\tag{1}$$
where $W(x)$ denotes the main branch of the Lambert $W$ function, as reference I add that Wikipedia ...
3
votes
1
answer
340
views
Proof of continued fraction identity of subfactorial
This question is part of a wider conjecture I have formed with someone which has its roots in Raayoni et al. (2019). The subfactorial function can be written as $$!n=\frac{n!}e+\frac{(-1)^n}{n+2-\...
- 1,459
5
votes
1
answer
232
views
"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 ...
- 51
1
vote
2
answers
259
views
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,\...
- 798
1
vote
1
answer
199
views
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}/\...
- 798
0
votes
0
answers
234
views
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 ...
- 3,094
2
votes
2
answers
268
views
Growth rate for the average of the entries in the fundamental period of the continued fraction expansion of $\sqrt{n}$
(Cross-posted from stackexchange: https://math.stackexchange.com/questions/1976296/what-is-known-about-the-average-of-the-partial-quotients-in-the-fundamental-peri)
I'd like references concerning ...
- 1,521
2
votes
0
answers
276
views
Identity with Ramanujan's generalized continued fraction
Let $F(x,q)=\sum_{n\geq 0}x^n\dfrac{q^{n^2}}{(q)_n}$, where $(q)_n=(1-q)(1-q^2)\dots(1-q^n)$. Then:
$$H(x,q)=\frac{F(-xq,q)}{F(-x,q)}=\dfrac{1}{1-\dfrac{qx}{1-\dfrac{q^2x}{1-\dots}}}$$ is the ...
- 301
10
votes
3
answers
566
views
Continued Fractions from Digit Streams
let $x=\sum_{i=1}^{\infty}\delta_i2^{-i},\ \delta_i\in\{0,1\}$.
Is there an algorithm that converts the sequence $(\delta_0,\ \delta_1,\ ...)$ of the binary digits of $x$ to the sequence $[a_0;a_1,\...
- 13.2k
4
votes
2
answers
209
views
Expected Cardinality of the First n Coefficients of a Continued Fraction
Is there anything known about the asymptotic expected growth of $card\{a_0,a_1,...,a_n\}$, where the $a_i$ are the first $n$ coefficient of a continued fraction $[a_0; a_1, ... ]$?
- 13.2k
2
votes
0
answers
116
views
Name of a difference of continuants
I am getting ready to publish the manuscript
http://arxiv.org/pdf/1408.4631v2.pdf
and I am trying to do due diligence on a quantity I study before it gets published. (This is cross-posted from Name/...
- 1,521
31
votes
3
answers
5k
views
Is any particular algebraic number known to have unbounded continued fraction coefficients?
The continued fraction
$$[1;1,2,3,4,5,\dots]=1+\cfrac{1}{1+\cfrac{1}{2+\cdots}}, $$ for instance, is known explicitly as a ratio of Bessel function values and is (I believe - SS) known to be ...
- 3,094
5
votes
2
answers
619
views
Are there any patterns in simple continued fraction expansions of algebraic real numbers?
As we know there are patterns in simple continued fraction expansion of quadratic algebraic numbers,are there any patterns in simple continued fraction expansions of other algebraic real numbers?...
- 3,094
7
votes
1
answer
856
views
Continued fraction representation of Zeta
A question at math.SE is asking for references. The fraction is quite nice! Check it out and post some references if you know of any.
I found this at arxiv, but it doesn't apply to Zeta.
- 1,043
10
votes
3
answers
638
views
Last term of repeating continued fraction expansion
Once again, working with stable vector bundles on $\mathbb{P}^2$ I have run into a question that is really out of my area. (Thanks to everybody who helped out with my last question!)
Let $D>9$ be ...
- 5,857
7
votes
3
answers
510
views
Proto-Euclidean algorithm
Consider the Euclidean algorithm (EA) as a way to measure the relative length $b/a$ of a shorter stick $b$ compared to a longer one $a$ by recursively determining
$$q_i = \left\lfloor \frac{r_i}{r_{...
- 9,720
17
votes
2
answers
3k
views
Some unpublished notes of Hofstadter
I'm looking for some unpublished notes called "Eta Lore," which are apparently related to a talk Douglas Hofstadter first gave at the Stanford Math Club in 1963. I know these notes exist because they'...
7
votes
2
answers
2k
views
Applications of periodic continued fractions
Some answers from Applications of finite continued fractions in fact are Applications of periodic continued fractions. I think that it should be separate question.
What can you add to the following ...
34
votes
21
answers
11k
views
Applications of finite continued fractions
I know some applications of finite continued fractions. Probably you know more. Can you add anything? (For Applications of periodic continued fractions I have made a special topic.)
1) (Trivial) ...
7
votes
2
answers
929
views
English translation of Voronoi's dissertation
I am looking for an English translation of Voronoi's doctoral dissertation, "On a generalization of the Algorithm of Continued Fractions." I can only find it in the original Russian.
- 173