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Questions tagged [irrationality-measure]

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0
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1answer
47 views

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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0answers
49 views

Can this product formula for $\pi$ be used to 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}\...
8
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1answer
126 views

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 ...
0
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1answer
26 views

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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2answers
33 views

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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0answers
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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=...
21
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1answer
623 views

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 ...
2
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0answers
38 views

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 ...
2
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1answer
37 views

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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1answer
77 views

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, ...
19
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1answer
1k views

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 ...
8
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1answer
382 views

Recent progress in the irrationality measure of $\pi$

The context. 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<...