Minkowski type functions on probability distributions
Symon Serbenyuk
TL;DR
We address constructing Minkowski-type functions on numbers represented by probability-distribution expansions. The approach defines $x=\pi_P((i_k))$ for a distribution $P=(p_i)$ and introduces the Minkowski-type map $M_\pi(x)=\sum_{k=1}^{\infty}(-1)^{k-1}2^{1-i_1-\cdots-i_k}$, establishing its continuity, nowhere monotonicity, and singularity. A shift-based functional equation $h(\sigma^{n-1}(x))=\frac{1}{2^{i_n}}(1-h(\sigma^n(x)))$ yields a self-affine graph $\Gamma=\bigcup_{t=1}^{\infty} \psi_t(\Gamma)$ and a closed-form Lebesgue integral $\int_0^1 M_\pi(x)\,dx=\frac{2\alpha}{1+\gamma}$ with $\alpha=\sum_{j\ge1} \frac{p_j}{2^j}$ and $\gamma=\sum_{j\ge1} \frac{p_j^2}{2^j}$. The work thereby links distribution-induced expansions to fractal singular functions and provides a framework for approximation-theoretic and numeral-system analyses based on probabilistic representations.
Abstract
In this research, Minkowski type functions which are constructed on certain probability distributions, are introduced. There are investigated differential, integral, and other properties of these functions.