G-Representation of Real Numbers and some of its Applications

We study a two-symbol numeral system with two bases with different signs ${g}_{0}=\frac{1}{2}$ and ${g}_{1}=-\frac{1}{2}$, which is an analog of the binary and nega-binary numeral systems. We prove that the new system also has zero redundancy, i.e., any number $x\in \left[0;\frac{1}{2}\right]$ has at most two expansions in series

$$x={g}_{{\alpha }_{1}}+\sum_{k=2}^{\infty }\left({\alpha }_{k}{g}_{1-{\alpha }_{k}}\prod_{i=1}^{k-1}{g}_{{\alpha }_{1}}\right)={g}_{{\alpha }_{1}}+\sum_{k=2}^{\infty }\left(\frac{{\alpha }_{k}}{{\left(-2\right)}^{{\alpha }_{1}+...+{\alpha }_{k}}{2}^{k-\left({\alpha }_{1}+...+{\alpha }_{k}\right)}}\right)={\Delta }_{{\alpha }_{1}{\alpha }_{n}\dots }^{G},$$

where α_k ∈ A = {0; 1}, moreover, there exists only a countable everywhere dense set (of so-called G-binary) numbers in $\left[0;\frac{1}{2}\right]$, which have two expansions ${\Delta }_{{\alpha }_{1}\dots {\alpha }_{n}01\left(0\right)}^{G}={\Delta }_{{\alpha }_{1}\dots {\alpha }_{n}11\left(0\right)}^{G}$.

We describe the geometry of this representation (the properties of cylindrical and tail sets) and establish the relationship with the classical binary representation, i.e., propose the formulas for transitions from one representation to another. We study the structural, variational, integral, and fractal properties of the projector of the G-representation of numbers into the classical binary representation, i.e., the function defined by the equality $p\left(x={\Delta }_{{\alpha }_{1}{\alpha }_{2}\dots {\alpha }_{n}\dots }^{G}\right)={\Delta }_{{0\alpha }_{1}{\alpha }_{2}\dots {\alpha }_{n}\dots }^{2}$. It is shown that the projection continuous at G-unary points and discontinuous at G-binary points is a function of unbounded variation and has a self-similar graph.