# Approximate sequence of numbers

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 $$C_3=1$$.