The Engel--Minkowski question-mark function
Symon Serbenyuk
TL;DR
The paper defines an Engel-series based analogue of Minkowski's question-mark function, denoted $?_{EM}$, using Engel expansion digits $a_k$ via the representation $x = \sum_{k} \prod_{m=1}^{k} \frac{1}{2 + a_1 + \dots + a_m - m}$ and the alternating dyadic-like expansion $?_{EM}(x)/2 = 2^{-a_1} - 2^{-a_1-a_2} + \cdots$. It establishes continuity on $(0,1)$, proves that $?_{EM}$ is nowhere monotone, and derives a formula for its derivative as a limiting product, showing it can be $0$ or $\infty$ depending on the Engel digits. The work frames a functional-equation system with shift $\sigma$, $f(\sigma^{n-1}(x)) = 1/2^{a_n} - (1/2^{a_n}) f(\sigma^n(x))$, proves a unique bounded solution, and provides a self-affine/ cylinder-based analysis along with a closed-form Lebesgue integral. This establishes a foundation for Engel-series–driven Minkowski-type fractal analysis and potential applications to number representations and numeral systems.
Abstract
The present article deals with properties of a certain function of the Minkowski type with arguments defined by Engel series. Differential, integral, and other properties of the function were considered.