# Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

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### What did Thue prove when, concerning rational approximations?

A follow-up to When did Liouville come up with the first transcendental numbers? In 1909, Thue showed that if $\alpha\in\mathbb R$ is algebraic of degree $n$ and $s>\frac12n+1$, and if $c$ is any ...
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### Pseudo-lonely runner conjecture with $\frac{1}{k+1}$ and generalizations

I was reading about the unsolved lonely runner conjecture on Wikipedia, which states "[c]onsider $k$ runners on a circular track of unit length. At $t=0$, all runners are at the same position and ...
138 views

### Does $\min |{\cos(n)}|$ exist?

Let $f:\mathbb{Z^+}\rightarrow \mathbb{R}$ where $f(n)=|\cos(n)|$ in radians. Does $\min f$ exist? I think the answer is no and I that have the right approach to proving it. From Dirichlet ...
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### Dirichlet approximation $\frac{p}{q}$ s.t. $\left| \alpha-\frac{p}{q}\right| < \frac{1}{qN}$ where $N^{\epsilon} <q <N^{1-\epsilon}$

Dirichlet's Approximation Theorem says that for all $\alpha$ and $N$ there exists $q\in [N]$ and $p$ s.t. $$\left| \alpha-\frac{p}{q}\right| < \frac{1}{qN}.$$ This is a straighforward ...
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### Chromatic polynomial of the cross-polytope and denominators of convergents to e.

Let $C_n$ denote the $1$-skeleton of the $n$-dimensional cross-polytope, and $\chi_{C_n}(x)$ be the chromatic polynomial of $C_n$. This is equivalent to the way of coloring the $(n-1)$-dimensional ...
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