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I was looking for a cool thing to visualize, when I found this picture on Wolfram MathWorld. I obtained the image below by taking all integers $a, b, c, d$ in a given range (between $-15$ and $15$ here). I then calculated $\frac{a+bi}{c+di}$ and plotted it to give this image.

enter image description here

If I'm correct, these complex irreducible rationals are called the "Gaussian rationals". One thing that immediately pops out here are the empty regions with flower-like shapes.

It kind of reminds me of the Farey sequence, and it seems that the gaps here line up well with the gaps in the Farey sequences. There are gaps at $i$, $-i$, $1$, and $-1$, and smaller gaps at $\frac{1}{2}, \frac{i}{2}$, etc.

My question is - why? Why are there gaps at regular intervals in this picture, and why are there gaps in the Farey series?

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I think this is the Gaussian version of "rational numbers with small denominator can't be too close together". In the real rationals, this is just due to the inequality $|\frac wx-\frac yz| = \frac{|wz-xy|}{xz} \ge \frac1{xz}$ (when $\frac wx\ne\frac yz$ of course).

In the Gaussians, the distance between $\frac{a+bi}{c+di}$ and $\frac{e+fi}{g+hi}$ will be at least $\frac1{\sqrt{(c^2+d^2)(g^2+h^2)}}$. So there's a gap around each Gaussian rational (when comparing to other Gaussian rationals of small height), but those gaps are more pronounced when $c^2+d^2$ is small (such as $\pm1$ and $\pm i$).

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  • $\begingroup$ Okay, but do you have any papers or sources that confirm this claim? I want to read more if so. $\endgroup$
    – zenzicubic
    Commented Jan 12, 2023 at 23:01
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    $\begingroup$ You can confirm it yourself: \begin{align*} \biggl| \frac{a+bi}{c+di} - \frac{e+fi}{g+hi} \biggr| &= \biggl| \frac{(a+bi)(g+hi)-(e+fi)(c+di)}{(c+di)(g+hi)} \biggr| \\ &= \frac{|(-ce+df+ag-bh)+i(-de-cf+bg+ah)|}{|(c+di)(g+hi)|} \ge \frac1{|c+di||g+hi|} \end{align*} $\endgroup$ Commented Jan 13, 2023 at 1:25

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