So I read about this part from the textbook Illustrated Theory of Numbers (by Martin Weissman). The link to the part: https://illustratedtheoryofnumbers.wordpress.com/2012/07/31/fractions-and-semiotics/
Which states that 2 fractions $\frac{a}{b}$ and $\frac{c}{d}$ "kisses" if and only if $ad - bc = 1$ or $= -1$
Say we have $\frac{2}{3}$ , I can find $\frac{x}{y}$ that kisses $\frac{2}{3}$ by solving Diophantine equation $2y - 3x = 1$ or $2y - 3x = -1$.
By solving for $f(x,y) = 1$, I get a general formula for $\frac{x}{y}$ which is $\frac{2n+1}{3n+2}$ such that it will be smaller than $\frac{2}{3}$. And:
$\frac{2n+1}{3n+1}$ such that $\frac{x}{y}$ will be bigger than $\frac{2}{3}$ for the case of $f(x,y) = -1$ for all integers n.
Why is this the case?
The point is that $\frac ab-\frac cd=\frac{ad-bc}{bd}$. So, assuming $b$ and $d$ are positive, $\frac cd$ is smaller than $\frac ab$ if $ad-bc>0$ and larger is $ad-bc<0$.
If this is $\pm 1$ then it will be greater or less by the minimum possible amount, given the size of its denominator.