# Fractal covering of a plane with complex-base numeral systems - is periodicity necessary?

Taking a base $z$ positional numeral system with digits $a_k\in \{0,\ldots,n-1\}$:

$$s:\left\{(a_k)\in\{0,\ldots,n-1\}^{\mathbb{Z}}: \exists_K \forall_{k>K} \ a_k=0\right \}\to \sum_{k\in\mathbb{Z}} a_k z^k$$

we usually consider $z=n$ (or $z=-n$), making $s$ "nearly bijection" with $\mathbb{R}^+\cup\{0\}$ (or $\mathbb{R}$): surjection and injection on all but a zero measure set (there is countable number of values with double representation, e.g. $0.11111(1) = 1.00000(0)$ for binary system $n=z=2$).

It turns out that we can also get such nearly bijection (not injective only on zero measure set) in higher dimensions, requiring e.g. that $|z|^2=n$ in 2D or $||Z||^3=n$ in 3D. Here are examples of using first non-negative powers for complex base numeral systems (easy to draw on graph-ruled notebook): $n=2$, $z=1\pm i$, the right one further recreates $\mathbb{Z}^2$ lattice:

Fractional part: $F_z=\left\{\sum_{k<0}a_k z^k:\ a_k\in \{0,\ldots,n-1\}\right\}$ is a simple IFS fractal: fulfills $z F_z=F_z \cup (F_z+1)\cup\ldots\cup (F_z+n-1)$.

Integer part: $I_z=\left\{\sum_{k\geq 0}a_k z^k:\ a_k\in \{0,\ldots,n-1\right\},\ \exists_K \forall_{k>K}\ a_k=0\}$ fulfills $I_z =zI_z\cup(zI_z+1)\cup\ldots \cup (zI_z+n-1)$.

We would like surjectivity: $I_z+F_z=\mathbb{C}$ and injectivity on all but a zero measure set (boundaries of such fractals): $\mu(F_z \cap (F_z+1))=0$.

There remains a difficult question of choosing $z=\sqrt{n}e^{i\varphi}$ argument to get it. Working on it a long time ago, I have concluded (without proof) that we need $I_z=\mathbb{Z}z+\mathbb{Z}$, requiring:

$$\textrm{periodicity condition}: z^2 \in \mathbb{Z}z+\mathbb{Z}$$

There is infinite (countable) number of such cases (twindragon and tame-twindragon for $n=2$, more for higher $n$), and we can tell nearly everything about such fractals, including analytic formula for area of fractional part, Hausdorff dimension of its boundary, circumference and area of its convex hull - some my materials: paper, slides, 3 demonstrations, fractal Haar wavelats.

The remaining question is how to prove (or disprove) that periodicity is required for nearly bijection (not injective on at most zero measure set)?

Here are such fractional parts for $n=2$, $z=\sqrt{2} e^{i\varphi}$ and growing $\varphi\in[\pi/2, \pi)$, below are numerical results of their Hausdorff dimension - for surjectivity it has to be two. We can see the three marked surjective cases: rectangle, tame twindragon and twindragon. They are all periodic - the question is if it is true for any $n$? In other words: if there exists a non-periodic surjective case with $|z|^2=n$?

Simple Mathematica source to generate such fractals, $n$ and $\varphi$ as above, $d$ first digits:

d = 14; n = 2; phi = Pi/4.; z = Sqrt[n] Exp[I*phi]; pow = Table[ReIm[z^k], {k, 0, d - 1}];
Graphics[{Point[Table[IntegerDigits[x, n, d].pow, {x, 0, n^d - 1}]]}]