For questions involving Liouville numbers.
An irrational number $x$ is a Liouville number if for all $n \in \mathbb{N}$, there exist $p,q \in \mathbb{N}$ with $q>1$ such that $0<|x-\frac{p}{q}|<\frac{1}{q^n}$.
Questions tagged [liouville-numbers]
28 questions
14
votes
1 answer
Liouville numbers and continued fractions
First, let me summarize continued fractions and Liouville numbers.
Continued fractions.
We can represent each irrational number as a (simple) continued fraction by $$[a_0;a_1,a_2,\cdots\ ]=a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{\ddots}}}$$
where…
user304329
7
votes
1 answer
Intuition behind the irrationality measure
The irrationality measure $\mu(x)$ of a real number $x$ is defined to be the supremum of the set of real numbers $\mu$ such that the inequalities
$$0 < \left| x - \frac{p}{q} \right| < \frac{1}{q^\mu} \qquad (1)$$
hold for an infinite number of…
tparker
- 6,756
7
votes
2 answers
Why is $e$ not a Liouville number?
A Liouville number is an irrational number $x$ with the property that, for every positive integer $n$, there exist integers $p$ and $q$ with $q > 1$ and such that $0 < \mid x - \frac{p}{q} \mid < \frac{1}{q^n} $.
I'm looking for either hints or a…
cdwe
- 671
5
votes
0 answers
Reciprocal of a Liouville number is also a Liouville number
Prove that the reciprocal of a Liouville number is also a Liouville number
I am using the definition of a Liouville number given in the book Transcendental Numbers by M. Ram Murty. A screenshot of the definition is attached
Proof: Let, if…
James Moriarty
- 6,460
- 1
- 20
- 39
5
votes
1 answer
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{equation}
In their 1982 paper on almost-periodic…
Eduard Tetzlaff
- 123
4
votes
2 answers
Proving the Liouville Numbers are uncountable
The definition I'm given for the Liouville numbers is just that $x$ is Liouville if for every $N > 0$ there exists integers $p,q \geq 2$ such that
$$\left|x-\frac{p}{q}\right| < \frac{1}{q^N}.$$
I don't have too many high powered theorems to work…
bears
- 113
4
votes
1 answer
Dirichlet's approximation theorem with even or odd denominators
It follows from Dirichlet's approximation theorem that for any irrational $\alpha,\ 0<\left\lvert \frac{p}{q} - \alpha \right\rvert < \frac{1}{q^2} $ for infinitely many pairs of integers $(p,q).$ In fact, Hurwitz's theorem is a nice result which…
Adam Rubinson
- 24,300
4
votes
1 answer
An unclear connection between two (supposedly) equivalent formulations of Liouville's approximation theorem
We shall say that $\xi\in\mathbb{R}$ is approximable by rationals to order n if there is a $K(\xi)$ for which $$|\frac pq -\xi|<\frac {K(\xi)}{q^n}$$ has infinitly many solutions. (according to the book: THE THEORY OF NUMBERS by Hardy &…
S.H.
- 81
3
votes
1 answer
Discussion questions involving Liouville numbers
I'm working through Measure and Category by Oxtoby with some friends, but since the book doesn't have any exercises, we needed to come up with our own to discuss. My friend found the following problems on the Baire Category theorem (from Rudin and…
user596778
3
votes
0 answers
Topology of Liouville Numbers
Liouville numbers are defined as the numbers $x\in \mathbb R$ s.t. for ever $i\in\mathbb N$ there exist $n,m\in\mathbb Z$ such that
\begin{equation}
0<\left|x-\frac nm\right|<\frac1{m^i}.
\end{equation}
and they have the property of being well…
Davide Maran
- 1,259
2
votes
1 answer
Finding an $a$ to obtain a given radius of convergence for $\sum \frac{z^n}{\sin(an\pi)}$
I have been trying to solve the following problem : prove that, for any $R\in [0,1]$, there exists an $a\in \mathbb R\backslash \mathbb Q$ such that the radius of convergence of $\sum \frac{z^n}{\sin(an\pi)}$ is $R$.
It turns out that when a is…
Tutut
- 21
2
votes
1 answer
Proof of Liouville's theorem and the construction of transcendental numbers - Hardy et al.
I've been trying to understand the proof of Liouville's theorem given in "Intro to Theory of Numbers"- G.H. Hardy et al. and not able to understand few steps. The proof snippet is attached.
Can someone please help me understand the following
a) How…
Resting Platypus
- 125
2
votes
2 answers
Algebraic number to Liouville number
If $a\in\mathbb{R}\setminus\left\{0,1\right\}$ is an algebraic number, can $\ln\left(a\right)$ ever be a Liouville number?
This is not a homework question, nor do I know much about the innards of proving these kinds of things. I am just very…
user301661
2
votes
0 answers
Prove that the set $S$ is equal to the set of all Liouville numbers
I need to prove that the set
$$S=\bigcap_{n=1}^\infty\bigcup_{q=2}^\infty\bigcup_{p=-\infty}^\infty\left(\frac{p}{q}-\frac{1}{q^n},\frac{p}{q}+\frac{1}{q^n}\right)-\left\{\frac{p}{q}\right\}$$
Is equal to the set $L$ of all Liouville numbers. It's…
Kash
- 139
2
votes
1 answer
$f(z)$ be an entire function there is a $C >0$ such that $|\Im f(z)| \le C$.
I have a question, if $f(z)$ is a entire function such that there exists a $C > 0$ such that $|\Im f(z)| \le C$. Show that $f(z)$ is a constant function. Have i made a mistake
I said let
$$
g(z)= e^z
$$
then
$$
e^{\Re f(z)}e^{\Im f(z)}|=|e^{\Im…
Sarah Angel
- 81