# Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

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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 ...
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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?
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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 ...
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### 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: ...
1 vote
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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. ...
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### 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 $p$,...
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 ...
1 vote
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### 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), ...
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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\$ ...
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Let $\varphi$ be the Euler totient function and $f$ be a nonnegative function such that $$\sum_{n=1}^{\infty} \frac{f(n)\varphi(n)}{n}=\infty,$$ and that $$\limsup_{N\to \infty} \frac{\sum_{n=1}^N f(n)... 1 vote 0 answers 38 views ### Measure or irrationality and discriminant For an algebraic number \alpha of degree 2, I know how to prove that there is a constant c>0 such that$$\left|\alpha - \frac{p}{q}\right| \geq \frac{c}{q^2}.$$However, I get the constant c... 1 vote 0 answers 52 views ### How to approxmate a real number by a sum of two real with integer weights? Given three real numbers, a, b, and c, I am wondering how to approximate c by the sum an+bm for integers n and m up to the desired accuracy. More precisely, define z:\mathbb{Z}\times \... 3 votes 1 answer 67 views ### What did Thue prove when, concerning rational approximations? A follow-up to When did Liouville come up with the first transcendental numbers? In 1909, Thue showed that if \alpha\in\mathbb R is algebraic of degree n and s>\frac12n+1, and if c is any ... 2 votes 1 answer 63 views ### Pseudo-lonely runner conjecture with \frac{1}{k+1} and generalizations I was reading about the unsolved lonely runner conjecture on Wikipedia, which states "[c]onsider k runners on a circular track of unit length. At t=0, all runners are at the same position and ... 4 votes 3 answers 157 views ### Does \min |{\cos(n)}| exist? Let f:\mathbb{Z^+}\rightarrow \mathbb{R} where f(n)=|\cos(n)| in radians. Does \min f exist? I think the answer is no and I that have the right approach to proving it. From Dirichlet ... 5 votes 0 answers 35 views ### Chromatic polynomial of the cross-polytope and denominators of convergents to e. Let C_n denote the 1-skeleton of the n-dimensional cross-polytope, and \chi_{C_n}(x) be the chromatic polynomial of C_n. This is equivalent to the way of coloring the (n-1)-dimensional ... 2 votes 1 answer 96 views ### Finding an integer vector perpendicular to a rational vector/solving a Diophantine equation with constraints. Essentially, I want to solve the following problem: Given a vector of rational numbers, \vec{X}\in \mathbb{Q}^d, find the smallest integer vector perpendicular to it. That is, find \vec{n}\in \... 3 votes 0 answers 49 views ### How are almost integers of the form (n^2-1)\pi related to convergents? Two fractions related to early convergents of \pi are the semi-convergents$$\frac{333-22}{106-7}=\frac{311}{99}$$and$$\frac{355+22}{113+7}=\frac{377}{120}$$The denominators of these fractions ... 10 votes 1 answer 294 views ### 355/113 and small odd cubes An important approximation to \pi is given by the convergent \frac{355}{113}. The numerator and the denominator of this fraction are at the same distance of small consecutive odd cubes.$$\frac{...
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{ ...
### 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 ...