# Tagged Questions

For questions about approximating real numbers by rational numbers.

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### Bound to the Number of Solutions of the Thue-Siegel-Roth Theorem

I have to prove for an assignment that if the partial quotients of a number $x$ are $\frac{p_n}{q_n}$, and $$\limsup\frac{\log\log (q_n)\sqrt{\log n}}{n}=\infty$$ then $x$ is transcendental. I do not ...
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### Rigorous rationale for the Pade Approximant?

I recently asked a Question for which the only Answer I got was a recommendation to punt and use the Pade Approximant. This is the first time I recalling seeing this, and intuitively it seems like ...
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### approximate a vector of complex numbers

Given a vector of complex number $\vec{z}=(z_1,\cdots, z_n)$ with $|z_i|=1$ and $z_i$ is not a root of unit, and a vector of complex numbers $\vec{r}=(r_1, \cdots, r_n)$ with $|r_i|=1$. Is it the case ...
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### Thue-Siegel-Roth Type Theorem

Dirichlet's approximation theorem says that for every real $\alpha$ and every positive integer $N$, there exist integers $p,q$ with $1 \leq q \leq N$ such that $$|q\alpha - p| < \frac{1}{N}.$$ ...
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### Yet another diophantine approximation question

Let $x=(x_1,\ldots,x_n)$ be a real vector in general position and let's say it is normalized: $\lVert x\rVert=1$. Let $y$ be a real number that can be arbitrarily large and $\epsilon>0$ can be ...
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### An integral for $2\pi+e-9$
Motivation Lucian asked about the almost-integer $2\pi+e\approx9$ in a comment to a partially answered why question about $e\approx H_8$. This is more involved than approximations to $\pi$ and ...