# Questions tagged [irrationality-measure]

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12 questions
1answer
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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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49 views

1answer
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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 ...
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 ...
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 ...
1answer
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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, ...
1answer
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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 ...
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<...