What fraction of fractions does Cantor’s famous sequence enumerate?

Cantor’s famous sequence 11,12,21,13,31,14,23,32,41,15,51,16,… provides a bijection between natural numbers and positive rational numbers or cancelled fractions. About half of the fractions qi lie within 0<x≤1. What is the limit of the ratio limk→∞|{x∈R|n<x≤n+1}∩{q1,q2,…,qk}||{x∈R|0<x≤1}∩{q1,q2,…,qk}| for n \in \mathbb{N}? Is there an n for which the limit is 0? And if so, what is the first … Read more

Numerical coincidence?

(Nobody’s answered this one on stackexchange after several days.) My brother built a garage whose horizontal cross-section is a rectangle that measures $45$ feet by $30$ feet. To make sure the right angles were accurate, he measured the two diagonals of the rectangle to see that they were equal. In inches, \begin{align} & \sqrt{540^2+360^2} \approx … Read more

Liouville’s Theorem in Diophantine Approximation

Liouville’s Theorem states that for any algebraic $\alpha \in \mathbb{R}$ of degree $n$, there exists a positive constant $c:=c(\alpha)$ such that $$\left\lvert\alpha-\frac{p}{q}\right\rvert>\frac{c}{q^n}$$ for any $p \in \mathbb{Z}$ and $q \in \mathbb{N}.$ One can find an effective lower bound for $c(\alpha).$ In the special case that $\alpha$ is a quadratic irrational, Exercise 27 in the following … Read more

Mean value of a function associated with continued fractions

Suppose that an irrational $x$ in $(0,1)$ has convergents $c(k,x)$, and let $$d(x) = \sum_{k=0}^{\infty} \mid x – c(k,x)\mid.$$ What is the mean value of $d$? Answer If $\frac{p_{2k}}{ q_{2k}}$ and $\frac{p_{2k+1}}{ q_{2k+1}}$ ($k\ge 0$) are consecutive convergents of the continued fraction expansion of $x$ then $$\left|x-\frac{p_{2k}}{ q_{2k}}\right|+\left|x-\frac{p_{2k+1}}{ q_{2k+1}}\right|=\frac{p_{2k+1}}{ q_{2k+1}}-\frac{p_{2k}}{ q_{2k}}=\frac{1}{q_{2k}q_{2k+1} }.$$ For example if … Read more

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 $[a_0; a_1, … ]$? Answer Since the question asks about an expectation, one needs to specify a probability space of continued fractions. It seems the most natural such space is the interval … Read more

A special class of weighted Motzkin paths

Consider Motzkin paths with the following weight: All up-steps and the horizontal steps on height $0$ have weight $1$, all down-steps have weight $t$ and the horizontal steps on even heights have weight $1-t$ while those on odd heights have weight $t-1.$ Computations suggest that the weight $c_n(t)$ of all paths of length $n$ is … Read more

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 fraction expansions of $x$ and $x’$ have the same period? Computations of a few random example seems to suggest that is indeed … Read more

Generators of a 2D lattice

Dear MO_World, I’m hoping someone can point me towards a reference for something. I have an invertible $2\times 2$ matrix, $A$, with real entries such that for both of the rows, the entries are rationally independent (this ensures that $A\mathbb Z^2$ only intersects the coordinate axes at the origin). What I want is a pair … Read more

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?Or any law in them?or is there any universal algorithm to compute the integer sequence in simple continued fraction expansions,with the equation whose one solution is the real … Read more

“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, x2−dy2=−1 has no solution. Then we know the minimal period of the continued fraction expansion of √d has even length, 2ℓ, and that the partial denominators, a1,…,a2ℓ−1, are symmetric about the ℓ-th partial denominator, aℓ. I.e., aℓ−j=aℓ+j … Read more