Hausdorff dimension for the weighted products of multiple digits in d-decaying Gauss like systems

TL;DR

This paper calculates the Hausdorff dimension of sets formed by weighted products of multiple digits in d-decaying Gauss-like systems, extending previous results to more digits without requiring Bounded Distortion Property.

Contribution

It provides the first complete Hausdorff dimension results for products of more than two digits in these systems, solving an open problem.

Findings

Derived explicit Hausdorff dimension formulas for multiple digits

Extended results to arbitrary digit positions in d-decaying systems

Removed the need for Bounded Distortion Property in analysis

Abstract

We compute the Hausdorff dimension of sets defined by the growth of weighted products of multiple digits at arbitrary positions in $d$-decaying Gauss-like iterated function systems. We provide the complete Hausdorff dimensional result for product of more than two digits, which was an open problem even for consecutive digits in the classical Gauss map and L\"{u}roth map. In our approach we do not need to assume the Bounded Distortion Property (BDP).