# Tagged Questions

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### Representing sparse set as set of extremely good approximation

For $\alpha \in \mathbb R$ and $\varepsilon(n) > 0$, consider the set $$N(\alpha,\varepsilon) = \{ n \in \mathbb{N} \ : \ \lVert \alpha n \rVert < \varepsilon(n) \}$$ (where $\lVert x \rVert$...
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### On property of monic polynomial with integer coefficients

For a monic polynomial with integer coefficients $f$ where $\partial f = 2$, we have $$\textrm{inf}(f(x)) > 0 \implies \textrm{inf}(f(x)) \geq \frac{3}{4} .$$ Could we generalize this (for ...
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### On a sequence of integers

I recall a well-known theorem due to Minkowski. Theorem. If $\theta$ is irrational and $\alpha$ is not of the form $\alpha = m\theta+n$ for some integers $m$ and $n$, then there are infinitely many ...
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### On cluster points of a particular sequence

This is the sequel of a previous question. Let us consider the sequence $$\xi_n = 2n \{n\xi\}-n,$$ where $\xi>0$ is a given real irrational number and $\{\cdot\}$ is the fractional part. Do ...
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### Precise asymptotic of diophantine approximation

I know that every irrational number $\xi$ can be approximated by rational numbers in such a way that $$\left| \xi - \frac{p}{q} \right| \leq \frac{c}{q^2}$$ for infinitely many choices of $p$ and $q$...
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### A particular Diophantine approximation of $\pi/2$

I have asked this question in math.stackexchange without any answer, so I have decided to post it here too. Recently I was playing around with the sequence $$\frac{1}{n\sin(n)},\ n\in\mathbb{N}.$$ ...
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### Rate of convergence of an irrational rotation

Let $\alpha, \beta \in \mathbb{R}$. Let $\{x\}$ denote the fractional part of $x$ and let $\|x\| = \min(\{x\}, 1-\{x\})$. If we assume that $\alpha$ is irrational, then there exists an increasing ...
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### Computing all “suboptimal” rational approximations to $\pi/2$

I have an irrational number $\alpha$ ($\alpha=\frac\pi2$), and I would like to determine all integers $n\in[1,N]$ ($N=10^{16}$) that satisfy $$n \epsilon(n)^2 \leq \tau$$ where $\tau$ is a known ...
Title question description: Select two lattices $\Lambda_1$ and $\Lambda_2$ (here a lattice=additive free abelian group without accumulation points) of maximal rank two in the real plane. We normalize ...