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### Analog of Baker's theorem on linear combination of $\log a \log b$

Baker's theorem basically says that, given algebraic numbers $a_1,\ldots,a_n$ and $m_1,\ldots,m_n$, if there is no good reason for a linear combination $$\sum m_i\log a_i$$ to cancel, then it is ...
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### Efficient Dirichlet approximation (continued fractions?) over a number field

Is there an efficient algorithm for Dirichlet approximation for a given (high-degree) number field and its ring of integers, perhaps analogous to the Euclidean/continued fractions algorithm for the ...
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### Optimal Diophantine approximation of $\pi$

If the 'optimal' Diophantine approximation of $\pi$ is given by the maximum value of $M=-\log_q(\min_{\forall p \in \mathbb{N}} |\frac{p}{q}-\pi|)$ for $q \geq 2$, what is the maximum value of $M$?
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### Criterion for irrational numbers of constant type 2

From Kuiper's and Niederreiter's book Uniform distribution of sequences, Ch.2, § 3, I learn that an irrational number $\alpha\in \mathbf{R}\smallsetminus \mathbf{Q}$ is of constant type $\eta$ if ...
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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 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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### On the density of the sequence $\{n \{n \xi \} \}_n$

I have a question that I can't manage to answer myself. It comes from some work in PDE theory, but it is related to analytic number theory. Let us say that we have an irrational number $\xi$. The ...
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### Looking for an effective irrationality measure of $\pi$

Most standard summaries of the literature on irrationality measure simply say, e.g., that $$\left| \pi - \frac{p}{q}\right| > \frac{1}{q^{7.6063}}$$ for all sufficiently large $q$, without giving ...
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### Small quotients of smooth numbers

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good ...
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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 ...
### 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 ...