So I have a research problem which states that we compute the probability mass function of the random variable which returns the smallest denominator of a reduced fraction in a randomly chosen real interval of radius $\frac{\delta}{2}$ and my advisor proved the expected value of the minimum denominator to be $\frac{16}{\pi^2 \sqrt{\delta}}$ as $\delta \to 0$. I am asked to extend this to the complex case where a random open ball of radius $\delta$ is chosen and we would like for the min expected value of the norm of a denominator of a randomly chosen Gaussian rational in the ball. I do not know where to even begin with this, any hints or tips as this is my first research project. I know $\Bbb{Q}^2$ is dense in $\Bbb{C}$ so there exists infinitely many Gaussian rationals in any arbitrary chosen ball of fixed radius. Any hints in the right direction greatly appreciated. Here is a link to the paper.
I've read the paper maybe $300$ times thus far and can. barely grasp the notion of Farrey fractions (which from my understanding is all reduced fractions between $0$ and $1$ such that the denominator is less than a given, fixed, natural number). Also, I am unfamiliar with big-oh notation which appears in the paper. Basically would like to know what changes need to be made to the case on $\Bbb{R}$ to swap the result so it holds for $\Bbb{C}$. I have been staring at this paper for weeks and have made zero progress.
