# 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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### 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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### 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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### 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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