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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 $1/n^k.$

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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 $C_3=1$.

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