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