Let $n \in \mathbb N$ and $k_n \in \left\{0,..,n \right\}$ then we define the numbers

$$x_{n,k_n} = \frac{k_n+n^2}{n^3+n^2}.$$

It is easy to see that these numbers satisfy

$$x_{n,0} = \frac{1}{n+1} \le x_{n,k_n} \le x_{n,n} =\frac{1}{n}.$$

I would like to know whether there exist three constants $C_1,C_2,C_3>0$ and an integer $i \in \mathbb N$ such that we can find for every $x_{n,k_n}$ a $reduced$ fraction $$\frac{p_{n,k_n}}{q_{n,k_n}}$$ such that two conditions hold:

1.) The denominator can be controlled nicely:

$$ \frac{C_1}{n^i} \le \frac{1}{q_{n,k_n}} \le \frac{C_2}{n^3}$$ and

2.) The approximation is sufficiently good:

$$\left\vert x_{n,k_n}-\frac{p_{n,k_n}}{q_{n,k_n}} \right\vert \le \frac{C_3}{n^3}.$$

So to summarize: I am wondering whether one can approximate the $x_{n,k_n}$ by reduced fractions up to an error of order $1/n^3$ and whether those fractions can have a denominator that is always between two different powers of $1/n^k.$


This is possible: Choose $q_n$ a prime number between $n^3$ and $2n^3$ (even independent of $k_n$) which exists by the Bertrand–Chebyshev theorem, and choose $p_{n,k_n}$ between $1$ and $q_n-1$ such that $\left| x_{n,k_n}-\frac{p_{n,k_n}}{q_n} \right|$ is minimal (in particular less than $\frac{1}{q_n}$). Then $\frac{p_{n,k_n}}{q_n}$ is obviously a reduced fraction, and your conditions are satisfied with $C_1=1/2$, $i=3$, $C_2=1$ and $C_3=1$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.