Questions tagged [irrationality-measure]

The Liouville-Roth irrationality measure of a real number $x$ is a measure of how "closely" it can be approximated by rationals.

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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)....
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A question about the irrationality measure formula

The irrationality measure $\mu(x)$ of a number $x\in\mathbb{R}\setminus\mathbb{Q}$ is defined as the inifimum of the set $$R_x=\{\mu>0: \text{for some}\, Q_\mu>0, \, \Big|x-\frac{p}{q}\Big|>\...
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If $A\subset \mathbb{N}$ is large, then does $\sum_{n\in A} \frac{\vert \sin n \vert }{n}$ diverge also?

If $A\subset \mathbb{N}$ is large, that is, $\displaystyle\sum_{n\in A} \frac{1}{n}$ diverges, then does $\displaystyle\sum_{n\in A} \frac{\vert \sin n \vert }{n}$ diverge also? I know that $\...
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Constructing an isosceles right triangle with $\sqrt{2}$ sides

Can you actually have an isosceles right triangle with $45$ degree angles? Such a triangle has sides of $2$, $\sqrt{2}$, and $\sqrt{2}$. But $\sqrt{2}$ is an irrational number. Can you actually have ...
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Irrationality measure of e

We define the irrationality measure of a number $x\in\mathbb{R}$ as the supremum of the $r\in\mathbb{R}$ such that exists $c\in \mathbb{R}$ verifying $$|x-\frac{p}{q}|\geq \frac{1}{q^r}$$ I was trying ...
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Strassmann's thoerem and irrationality measure of certain number

In this note from Keith Conrad, he explains an interesting application of Strassmann's theorem to the divergence of certain linear recurrence integer sequence. More precisely, the sequence defined as $...
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If $x,y>1,$ does there exist $\varepsilon>0,\ N,M\in\mathbb{N}$ s.t. $\lvert x^n-y^m\rvert>\varepsilon\quad\forall n\geq N,\quad \forall\ m\geq M?$

I was motivated by my previous question here to ask a more general question: Suppose $\ x,y\in\mathbb{R}_{>1}\ $ and $\ \not\exists\ p,q\in\mathbb{N}\ $ such that $\ x^p = y^ q.\ $ Then is the ...
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Given $\alpha\in \mathbb{R},$ do there exist arbitrarily large $m,k$ such that $\vert \alpha - \frac{k}{m}\vert < \frac{1}{m^2}$? [duplicate]

Given $\alpha\in \mathbb{R},\ n\in\mathbb{N}$ does there exist $m\in\mathbb{N}$ with $m>n,\ $ and $\ k\in\mathbb{Z}, $ such that $$ \alpha \in\left[\frac{k}{m}-\frac{1}{m^2}, \frac{k}{m} + \frac{1}{...
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Why does $|x-p/q|\ge\frac{C(x)}{q^a}$ for all $p/q$ imply: "$|x-p/q|\lt\frac{1}{q^a}$ has only finitely many solutions"?

This might be quite a trivial question but this has been bugging me, I don't know what the resolution is. Context: the irrationality measure $\mu=\mu(x)$ of some real number $x$ is defined to be that ...
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Showing that linear fractional equivalences preserve the irrationality measure

Throughout this post, let $(a,b,c,d)$ refer to the entries of some unspecified element: $$\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{GL}_2(\Bbb Z)$$By a “linear fractional equivalence” (...
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Understanding Liouville numbers and irrationality measure

Every number $x \in \mathbb{R}$ has an associated irrationality measure $\mu(x)$. Let $\mathbb{A}$ be the algebraic numbers and let $x_\mathbb{Q} \in \mathbb{Q}, x_{\mathbb{A}\setminus\mathbb{Q}} \in \...
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Let $a_1$ be linearly independent to $a_2$ over $\mathbb{Q}.$ For $n\geq 3,$ let $ a_n = \vert a_{n-1} - a_{n-2} \vert.$ Does $\sum_n a_n\ $ converge?

Let $a_1$ be linearly independent to $a_2$ over the rational numbers. For $n\geq 3,\ $ let $ a_n = \vert a_{n-1} - a_{n-2} \vert.$ Does $\sum_n a_n\ $ converge? For example, let $a_1 = 1,\ a_2 = \ln 2=...
Adam Rubinson's user avatar
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Why is irrationality mesure of irrational numbers at least 2

For any $x \in \mathbb{R}$, define its irrationality measure $\mu(x)$ to be the smallest number $\mu$ such that the inequality $|x-\dfrac{p}{q}| > \dfrac{1}{q^{\mu+\epsilon}}$ holds for any $\...
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Decimal expansions of $0.999\cdot\cdot$ and $1.000\cdot \cdot$ (infinite digits)

I am reading a passage from the book Foundation of Mathematics by Ian Stewart, and I need some help to make sure I understand it properly. A real number can be expresed by the following unique decimal ...
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Are Rational Power of e is transcendental?

It is well known to all of us that the rational powers of $e$ are irrational numbers. Many of the proofs proving this use a similar approach as proving $e$ irrational using Niven's Polynomials. Is it ...
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Measuring Irrationality à la 3Blue1Brown vs Liouville

So, I was watching this 3B1B video on measure theory and music (as you do) and was struck by how Grant uses epsilon to measure "how irrational" a real number is. He takes all the rationals $[...
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Approximation of a random number with quadratic integers

Consider the following claim: Claim: Let $x$ be a real random variable distributed according to the uniform distribution on the unit interval $U(0,1)$. Then for any quadratic irrational number $\...
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Roth's theorem: contradiction?

Roth's theorem says that for irrational algebraic number $\alpha$ and $\epsilon>0$, there are finitely many solutions to this: $$\displaystyle \left|\alpha-\frac pq\right|<\frac 1{q^{2+\epsilon}}...
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About irrationality measure.

The irrationality measure of $\alpha\in\mathbb{R}$ is defined by $\displaystyle\mu(\alpha)=\inf\left\{\nu\in\mathbb{R}_+,\; \text{card}\left(\left\{\frac pq\in\mathbb{Q},\; 0<\left|\alpha-\frac pq\...
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If |a - b| < |a - c|, then |1 - $\frac{a}{b}$| < |1 - $\frac{a}{c}$|?

If |a - b| < |a - c|, then |1 - $\frac{a}{b}$| < |1 - $\frac{a}{c}$| Is that true for all real values? If so, can someone point me to a proof? (EDIT: I made a mistake in my context. When ...
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Reference request about an irrationality measure problem

Some notations. For any real number $x$, let's define the quantity $$\mu(x):=\inf\left\{\mu\in\mathbb R_+\, \text{there is an infinity of rationals $p/q$ such that}\ \left\vert x-\frac pq\right\vert&...
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Convergence of a sine series involving Liouville's constant

I was investigating series whose convergence depends on a number's irrationality measure and would like to know if the sorts of questions I ask have been considered before and where I could look to ...
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What is the measure of $x^2$ in the space L2(-1,1)?

I have three sets of functions in $L_{2w}(-1,1)$ with w=1: $f_1 = x^2$ for all x $f_2 = x^2$ if x is irrational and zero otherwise $f_3 = x^2$ if x is rational and zero otherwise I want to know if ...
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On the proximity of $a\sqrt b+b\sqrt a$ to an integer

Let $\Sigma$ denote all ordered pairs $(a,b)$ of positive, square-free integers with $a> b$. What is the infimum of all $\kappa$ such that $$\left\{(a,b) \in \Sigma:a\sqrt b+b\sqrt a\,\,\text{is ...
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Are some irrational questions more irrational than others? (A-Level EPQ)

I'm currently researching the question 'Are some irrational questions more irrational than others?' for an extended project qualification I aim to write. My research has led me to seeing how spokes on ...
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Trying to calculate the top n most irrational numbers as continued fractions

I'm trying to do something outside my depth: determine the n-th most irrational number and display it in continued fraction form. I was told: The n-th most irrational number is equal to the n-th ...
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Is there a field of math dedicated to understanding irrational numbers

I have an interest irrational numbers and want to understand them. Is there a specific field for it?
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What about irrationality of $a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}}$?

This question is related to my question here which depend on the convergence of this sequence:$a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}}$ however the limit of convergence ...
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Do there exist numbers with non-integer irrationality measure?

Is it possible to construct a number (by way of an infinite series or a continued fraction say) having any, possibly non-integer, irrationality measure $>2$ ? It is known that this can be done for ...
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Applying the Beukers-like irrationality proof for $\zeta(2)$ and $\zeta(3)$ to Catalan's constant: where does it fail?

Many people have tried and failed to extend Apery's Irrationality proof of $\zeta(3)$ to Catalan's constant, by looking for a fast converging series for Catalan's constant analogous to the one for $\...
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Irrationality measure of a polynomial in $\pi$

Consider a number $x$ with known irrationality measure $r$ (for example $\pi$ with $7<r<8$). Is there anything we can say about the irrationality measure of a polynomial \begin{align} p=p(x)=\...
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Can the formula $\frac\pi2=(\frac21)^{1/2}(\frac{2^2}{1\cdot3})^{1/4}(\frac{2^3\cdot4}{1\cdot3^3})^{1/8}\cdots$ prove the irrationality of $\pi$?

A less known product formula for $\pi$, due to Sondow, is the following: $$ \frac{\pi}{2}= \left(\frac{2}{1}\right)^{1/2} \left(\frac{2^2}{1\cdot3}\right)^{1/4} \left(\frac{2^3\cdot4}{1\cdot3^3}\...
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Does $\sum_{k=1}^n|\cot \sqrt2\pi k|$ tends to $An\ln n$ as $n\to\infty$?

Question: How can we prove that $$L(n)=\sum_{k=1}^n\left|\cot \sqrt2\pi k\right|=\Theta(n\log n)$$ as $n\to\infty$? Furthermore, if $\sqrt2$ is replaced with a quadratic irrational number, does it ...
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Prove that irrationality measure is never less than 1

Every publication about the irrationality measure $\mu(\alpha)$ mentions as an obvious fact that $\mu(r) = 1$ for a rational $r$. Since I am new to this topic, it didn't look quite obvious to me, so ...
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If two real numbers are immesurable can an integer sum between the two get as close to any real number as we like?

Say for example we have $\pi$ and $1$. Can the sum $m\pi + n$ for $ m, n\in \mathbb{Z}^+$ get as close to a real number as we like? At first I tried using the fact that you could have $n = -floor(m \...
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Why can we not establish the irrationality of Catalan's constant the same way as $\zeta(3)$?

One of the main ingredients in Apéry's proof of the irrationality of $\zeta(3)$ is the existence of the fast-converging series: $$ {\displaystyle {\begin{aligned}\zeta (3)&={\frac {5}{2}}\sum _{k=...
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Are all transcendental numbers a zero of a power series?

So I came across the concept of extending the notion of irrationality to higher degree polynomials. The base case of this is standard irrationality. That is, a number is irrational if it cannot be ...
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Can we make the definition of irrationality measure ‘stronger’?

By definition, $$0<\left\lvert \frac{p}{q}-x\right\rvert<\frac1{q^{\mu(x)-\epsilon}}$$ has infinitely many solutions $(p,q)$ for every $\epsilon>0$. However, to prove a theorem in a ...
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Silly question about irrationally measure

The irrationality measure can be defined as: Let $x$ be a real number, and let $R$ be the set of positive real numbers $\mu$ for which $$0<|x-\frac{p}q|<\frac1{q^\mu}$$ has (at most) finitely ...
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The irrational of such constant [duplicate]

In this question, I needed to assume in my answer that $e^{e^{e^{79}}}$ is not an integer. Is there some standard result in number theory that applies to situations like this? After several years, ...
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Can multiples of two reals stay separated?

This question is inspired by my own answer to this question. For a real number $x > 0$, define $$ S(x) = \{\lfloor kx \rfloor \mid k \in \mathbb N\}. $$ Are there positive real numbers $x, y$ such ...
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Recent progress in the irrationality measure of $\pi$

The context. For any real number $x$, let's define the quantity $$\mu(x):=\sup\left\{\mu\in\mathbb R_+\, \text{there is an infinity of $(p,q)\in\mathbb Z\times\mathbb N$ such that}\ 0<\left\vert x-\...
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irrationality measure

I was reading that you can associate a measure to any given number giving you "how irrational" the given number is. I was wondering is there any irrationality measure that would tell you that the ...
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