Consider these two sets of Eisenstein integers.
SET 1 :
constructed by these rules :
a) any unit is in the set.
b) if $x$ is in the set, then so is $2 x + 7$.
c) if $x$ and $y$ are in the set then so is $xy$.
SET 2 :
constructed by these rules :
any unit is in the set.
if $a$ is in the set, then so is $2 a + 11$.
if $a$ and $b$ are in the set then so is $ab$.
Let $\pi_7(n)$ be the counting function for numbers in the set SET1 up to radius $n$. Let $\pi_{11}(n)$ be the counting function for numbers in the set SET2 up to radius $n$.
So we have counting functions and ofcourse the concept of density.
$\pi_7(n)$ could converge to a constant times $n^2$. or get less and less dense. At an exponential rate... or other.
Similar with $\pi_{11}(n)$
We might get fractal structures. Or not.
It might happen that patters eventually change. An apparant lowering density might grow. Or the other way around.
The growth rate and density of both sets are an interesting question. And the potential similarities between them.
How it visually looks is also interesting. Are both fractals ? Pseudofractals ?
All all trends exponential or not ?
Some will argue that SET1 has a limiting density of 1.
Some will probably argue that SET2 grows much faster because $2x + 11 < 2 x + 7$ , while others will expect about the same growth rate EVENTUALLY.
I would love to see arguments and plots. And proofs ofcourse but I assume those are hard.
I asked similar questions before for the ordinary integers :
Density of extended Mersenne numbers?
So it felt natural to wonder about higher dimensions and other UFD rings.
The Eisenstein integers are the densest in 2D with respect to radius from the origin. Also the Eisenstein primes are the densest in 2D with respect to radius from the origin. Both can be somewhat explained by the small norm and the high hexagonal symmetry ofcourse.
Therefore I picked them over the Gaussians or others. I think visually they will also be nicer.
I picked $+7$ and $+11$ for a few reasons ; not too large, not too small. And prime. I just guessed the interesting cases , maybe my guess is a bit off. But ofcourse all $2x + k$ are potentially interesting.
But we start with $+7$ and $+11$ here. We have to start somewhere. Although I do not mind seeing other $k$ plotted or mentioned.
I wonder if we get spirals or fractals and if so what kind.
