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Faithfulness and fractal (quasi-)equivalence principles for Perron, Engel, and Pierce expansions

Mykola Moroz

TL;DR

The paper develops three fractal principles that unify the fractal theory of Perron, Engel, and Pierce expansions by introducing faithful covering frameworks and dimension-preserving maps. The fractal equivalence principle shows that the Hausdorff dimension of sets defined via positive and alternating Perron digits is identical, enabling transfer of fractal results across Perron variants. The fractal quasi-equivalence principles connect the classical and modified Engel expansions and the Pierce expansion across Perron/traditional notations, under explicit summability conditions, to equate their fractal dimensions for broad families of digit-sequence constraints. Together, these results explain known analogies and yield new fractal properties, providing a systematic method to transfer Hausdorff-dimension results between different number expansions. The framework hinges on faithful coverings, enabling streamlined proofs and a unifying view of fractal properties across these generalized expansions.

Abstract

We establish several unifying principles that clarify the fractal properties of classical number expansions, which are generalized by the Perron expansions. In particular, we prove the fractal equivalence principle for the positive and alternating Perron expansions, the fractal quasi-equivalence principle for the classical and modified Engel expansions, and the fractal quasi-equivalence principle for the Pierce expansions in the Perron and traditional notations. These results explain several known analogies and show that the Hausdorff dimension of sets defined by one expansion often coincides with that for another. The proofs rely on faithful families of coverings, for which we refine previously known estimates. In addition to deriving a range of known theorems as direct corollaries of previous results, our approach yields new fractal properties of the Engel and Pierce expansions and provides a systematic framework for transferring Hausdorff dimension properties between different expansions.

Faithfulness and fractal (quasi-)equivalence principles for Perron, Engel, and Pierce expansions

TL;DR

The paper develops three fractal principles that unify the fractal theory of Perron, Engel, and Pierce expansions by introducing faithful covering frameworks and dimension-preserving maps. The fractal equivalence principle shows that the Hausdorff dimension of sets defined via positive and alternating Perron digits is identical, enabling transfer of fractal results across Perron variants. The fractal quasi-equivalence principles connect the classical and modified Engel expansions and the Pierce expansion across Perron/traditional notations, under explicit summability conditions, to equate their fractal dimensions for broad families of digit-sequence constraints. Together, these results explain known analogies and yield new fractal properties, providing a systematic method to transfer Hausdorff-dimension results between different number expansions. The framework hinges on faithful coverings, enabling streamlined proofs and a unifying view of fractal properties across these generalized expansions.

Abstract

We establish several unifying principles that clarify the fractal properties of classical number expansions, which are generalized by the Perron expansions. In particular, we prove the fractal equivalence principle for the positive and alternating Perron expansions, the fractal quasi-equivalence principle for the classical and modified Engel expansions, and the fractal quasi-equivalence principle for the Pierce expansions in the Perron and traditional notations. These results explain several known analogies and show that the Hausdorff dimension of sets defined by one expansion often coincides with that for another. The proofs rely on faithful families of coverings, for which we refine previously known estimates. In addition to deriving a range of known theorems as direct corollaries of previous results, our approach yields new fractal properties of the Engel and Pierce expansions and provides a systematic framework for transferring Hausdorff dimension properties between different expansions.

Paper Structure

This paper contains 12 sections, 27 theorems, 133 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Faithful families of coverings generated by Perron expansions
  4. Main results: new fractal principles for Perron, Engel, and Pierce expansions
  5. Fractal equivalence principle for the positive and alternating Perron expansions
  6. Fractal quasi-equivalence principle for the classical and modified Engel expansions
  7. Fractal quasi-equivalence principle for the Pierce expansion in Perron and traditional notations
  8. Explanation of known analogies via fractal principles
  9. Explanation of known analogies between the modified Engel and Pierce expansions via the fractal equivalence principle for the Perron expansions.
  10. Explanation of known analogies between the classical Engel and Pierce expansions via the fractal quasi-equivalence principles for the Engel and Pierce expansions.
  11. New analogies between the Engel and Pierce expansions
  12. Proofs of interval covering theorems for $\mathfrak{P}$ and $\mathfrak{P}^-$

Key Result

Proposition 2.5

Each $P$-cylinder has the form $(a,b]$. Each $P^-$-cylinder has the form $\hbox{$(a,b)\setminus IS^{P^-}$}$. The $P$-cylinder and the $P^-$-cylinder with the same base $c_1\ldots c_k$ have the same diameter, given by where $|\cdot|$ denotes the diameter of the set, $r_0=\varphi_0$, and $r_n=\varphi_n(c_1,\ldots,c_n)$ for all $n=1,\ldots,k-1$.

Figures (4)

  • Figure 1: An illustration of Lemma \ref{['lemma1']}
  • Figure 2: An illustration of Lemma \ref{['lemma2']}
  • Figure 3: An illustration of Lemma \ref{['lemmadovirchist3alt']}: the case $x_2\leq\sup\Delta^{P^-}_{c_1\ldots c_{k}(r_k+1)}$ with odd $k$
  • Figure 4: An illustration of Lemma \ref{['lemmadovirchist3alt']}: the case $\inf\Delta^{P^-}_{c_1\ldots c_{k}(r_k+n+1)}<x_2<\sup\Delta^{P^-}_{c_1\ldots c_{k}(r_k+n+1)}$ with odd $k$

Theorems & Definitions (45)

  • Definition 2.1: Moroz2025Moroz2024
  • Definition 2.2: Moroz2024
  • Definition 2.3: Moroz2025
  • Definition 2.4: Moroz2025Moroz2024
  • Proposition 2.5: Moroz2025Moroz2024
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • Corollary 3.5
  • ...and 35 more