Various fractal dimensions coincide on self-similar fractals as the logarithm of self-copies the fractal includes divided by the logarithm of the factor by which the copies are smaller than the original.
But if the set includes its own magnified copies, the fractal dimension seems to be negative.
Now take the set of integers. It is a self-similar fractal. It includes two its own copies (the even numbers and odd numbers) magnified by a factor of 2. This means the dimension of the set of integers is $\log(2)/\log(1/2)=-1$
Similarly, the dimension of the set of Gaussian integers seems to be $\log(4)/\log(1/2)=-2$.
While various manifolds can be assigned negative dimension this way (for instance, natural numbers are also self-similar), the role of n-sphere of negative dimension would play infinite in all directions equally-spaced lattice.
This hints us that the "radius" of the lattice is proportional to its lattice step. The set of integers thus is the unit -1-sphere.
Moreover, looking at the formula of the volume of n-sphere
$$V_n(R) = \displaystyle{\frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}}R^n$$
we can see that the volume of unit sphere would be $1/\pi$ while the radius of a spere whose volume is $1$ would be $1/\pi$. This corresponds to a $\pi$-periodic lattice. In other words, a $\pi$-periodic lattice has volume of exactly $1$.
The dependency of the radius and lattice step is as follows: $R=\frac{s}{\pi^2}$
Considering a unit sphere it seems the outer envelope (the surface of the sphere) would be formed by the prime numbers.
I wonder whether this interpretation valid and whether it was suggested before.