# Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

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• 6,801
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### Lagrange's best approximations law proof by contradiction doubt

In the proof of Lagrange's best approximations law it is stated that the system of equations $$xp_n+yp_{n+1}=p\\ xq_n+yq_{n+1}=q\\$$ where $\dfrac{p_n}{q_n}$ is the rational is a convergent of the ...
• 6,801
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### lonely runner conjecture for 3 runners

I am looking at the lonely runner conjecture and it seems that people often prove that the stationary runner is lonely for 3 runners. Why is it not considered when the other runners are lonely?
• 23
1 vote
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### Why is $|dx-c|<|qx-p|$ the condition for the best rational approximations of the second kind

The fraction $c/d$ is a best approximation of the second kind for a number $x$ if for every other fraction $p/q$ with $q<d$, $|dx-c|<|qx-p|$ Given a number $x$, its best rational approximations ...
• 6,801
62 views

### Is the convergence of these two series equivalent? (They come from Khinchin's theorem and the Duffin-Schaeffer conjecture.)

I am trying to wrap my head around two theorems of Diophantine approximation: Khinchin's theorem and the Duffin and Schaeffer conjecture. To the best of my understanding, here is what they say: ...
• 109
1 vote
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• 1,437
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• 395
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• 511
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• 364
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### Finding a pair $(a,b)\in \mathbb R^2$ such that $\inf_{m,n\in \mathbb Z}\left|(m+\sqrt{3}n+a)(\sqrt{2}m+n+b) \right|>0$

I am trying the find a pair $(a,b)\in \mathbb R^2$ such that $\inf_{m,n\in \mathbb Z}\left|(m+\sqrt{3}n+a)(\sqrt{2}m+n+b) \right|>0$ For $a=b=\frac{1}{2}$, I speculate that the infimum is nonzero. ...
• 7,161
166 views

### The lower bound of $|3^p - 2^q|$ - how to derive from Baker's theorem?

In his blog, Terence Tao discussed the lower bound of $\vert 3^p - 2^q \vert$ in the following corollary. Corollary 4 (Separation between powers of $2$ and powers of $3$) For any positive integers $p$,...
• 123
1 vote
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### Best rational approximations to a ratio of three numbers $x:y:z$

If I have a ratio of two positive real numbers $x:y,$ then I can find the "best" rational approximations to it by writing it as a continued fraction (e.g. by repeatedly removing the integer ...
• 111
1 vote
134 views

### Finding the square root of a polynomial with radical

I have two positive integers, $a$ and $b$, such that $$a^2=28b\sqrt{8b^2+1}+80b^2+5, \tag{\star}$$ and I’d like to find $a$ in terms of $b$ (including $\sqrt{8b^2+1}$, if necessary/appropriate), ...
• 7,216
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### Does there exist $\ n,m\in\mathbb{N}\$ such that $\ \lvert \left(\frac{3}{2}\right)^n - 2^m \rvert < \frac{1}{4}\$?

Does there exist $\ n,m\in\mathbb{N}\$ such that $\ \lvert \left(\frac{3}{2}\right)^n - 2^m \rvert < \frac{1}{4}\$ ? I have tried for the first few integers $\ n,m\$ up until $\ m\approx30\$ ...
• 12.2k
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• 4,914
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### Effective equidistribution for number with bounded irrationality measure

This question asked for bounds greater than $\omicron\left(n\right)$ on the error $$E_n=|T\cap\{1,2,\ldots\}|-\ell n.$$ where  \ell=\lim_{n\to\infty}\frac{|T\cap\{1,2,\ldots\}|}{n} \qquad\text{ ...
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### Convergent subsequences of $\{ n \alpha \bmod 1 \}$
It's well known that if $\alpha \not\in \mathbb{Q}$, then the sequence $\{n \alpha \bmod 1\}_{n \geq 0}$ is dense in the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$. Does every convergent subsequence ...