Intereting Posts

Algorithm for scrolling through different orbits in a permutation group
Does the logical equivalence of 2 statements imply their semantic equivalence in everyday language?
An integral involving Fresnel integrals $\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$
Find the limit of $x_n^3/n^2$ if $x_{n+1}=x_{n}+1/\sqrt{x_n}$
Irrationality of $ \frac{1}{\pi} \arccos{\frac{1}{\sqrt{n}}}$
Simplifying an Arctan equation
What does multiplication mean in probability theory?
Give an example that $\overline{A \cap B} \neq \overline{A} \cap \overline{B}$
$I^2$ does not retract into comb space
General Continued Fractions and Irrationality
Graph of $|x| + |y| = 1$
References for relations between and classification of different set systems?
Linear congruence
If $ X = \sqrt{Y_{1} Y_{2}} $, then find a multiple of $ X $ that is an unbiased estimator for $ \theta $.
$\lim_{n\to \infty }\cos (\pi\sqrt{n^{2}-n})$ – second battle

Is one of those claims about continued fractions true ?

$1)$ Almost all real numbers have a continued fraction representation with a bounded sequence of entries.

$2)$ Almost all real numbers have a continued fraction representation with an unbounded sequence of entries.

- Approximating $\arctan x$ for large $|x|$
- How to do a very long division: continued fraction for tan
- Convergence of a Harmonic Continued Fraction
- A pair of continued fractions that are algebraic numbers and related to $a^2+b^2=c^m$
- a new continued fraction for $\sqrt{2}$
- Bi-linear relation between two continued fractions

Intuitively I would expect that the unbounded sequence is the “typical” outcome. How can I determine the measure of both types of continued-fraction-sequences ?

- If $\int f=0$ and $f(x) \ge 0$ for all $x \in \mathbb{R}^d$, then $f=0$ a.e.
- Lebesgue integral question concerning orders of limit and integration
- Is there a function with infinite integral on every interval?
- Does $\mu_k(U \cap \mathbb{R}^k)=0$ for all affine embeddings of $\mathbb{R}^k$ in $\mathbb{R}^n$ imply $\mu_n(U)=0$?
- Rotation $x \to x+a \pmod 1$ of the circle is Ergodic if and only if $a$ is irrational
- How common are probability distributions with a finite variance?
- I want to understand uniform integrability in terms of Lebesgue integration
- Relationship between degrees of continued fractions
- Minimum of $|az_x-bz_y|$
- Continued fraction of $e^{-2\pi n}$

Almost all real number have continued fractions expansion which is not only unbounded, but distribute according to the Gauss-Kuzmin measure. This follows from the fact that the Gauss map is ergodic. See for example here. This result should also appear in every textbook about ergodic theory and\or continued fractions (for example “ergodic theory with a view towards Number Theory” by Einsiedler and Ward has a whole chapter on this).

- How to find all groups that have exactly 3 subgroups?
- Confused on a proof that $\langle X,1-Y\rangle$ is not principal in $\mathbb{Q}/\langle 1-X^2-Y^2\rangle$
- Detect Abnormal Points in Point Cloud
- No. of ways of selecting 3 people out of n people sitting in a circle such that no two are consecutive
- countably infinite union of countably infinite sets is countable
- Intuitive explanation for Derangement
- how do you solve $y''+2y'-3y=0$?
- combinatorics: The pigeonhole principle
- If $\gcd( a, b ) = 1$, then is it true to say $\gcd( ac, bc ) = c$?
- Puzzle : There are two lengths of rope …
- Proving trigonometric equation $\cos(36^\circ) – \cos(72^\circ) = 1/2$
- Why is the coordinate ring of a projective variety not determined by the isomorphism class of the variety?
- Uniqueness proof for $\forall A\in\mathcal{P}(U)\ \exists!B\in\mathcal{P}(U)\ \forall C\in\mathcal{P}(U)\ (C\setminus A=C\cap B)$
- Does $\int_{0}^{\infty}{\sin{(\pi{x^2})}\over \sinh{(\pi{x}})\tanh(x\pi)}\mathrm{d}x$ have a simple closed from?
- Coin problem with $6$ and $10$