Applications of certain probability distributions on positive integers

TL;DR

This work addresses generalizing Salem-type functions by encoding numbers through $\pi_{\mathfrak p}$-expansions guided by a probability distribution on $\mathbb N$ and a digit-relabelling map $\phi$ to produce $G: \pi_{\mathfrak p}((n_j)) \mapsto \pi_{\mathfrak o}((m_j))$. The authors establish a self-similar, shift-operator framework that yields a continuous bijection on $[0,1)$ which is not monotone, and they analyze derivative behavior via the ratio $o_{\phi(j)}/p_j$, demonstrating possible singular or infinite-derivative cases under suitable subsequence choices. A self-affine functional equation with a unique solution for $G$ is derived, along with an explicit Lebesgue integral formula $\int_0^1 G(x)\,dx=\frac{\sum_j \widehat{o_{\phi(j)}} p_j}{1-\sum_j o_{\phi(j)} p_j}$. The results illuminate a rich class of fractal-like, singular, and non-monotone maps constructed from probabilistic digit expansions, with potential connections to fractal analysis and modeling of complex real-world phenomena.

Abstract

The main goal of this research is to model and investigate generalizations of functions from [31]. Arguments of modeled functions are presented by the representation $π_{\mathfrak p}$ from [22].

Theorems & Definitions (10)

• Example 1
• Lemma 1
• proof
• Theorem 1
• proof
• Corollary 1
• Example 2
• Lemma 2
• Theorem 2
• proof