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

TL;DR

The paper determines the Hausdorff dimension of limsup sets defined by weighted products of digits in $d$-decaying Gauss-like IFSs, yielding a precise formula $\dim_H E = s_0$ with $s_0=\inf\{s: P(s)\le A(s)\log B\}$. It develops a distortion framework under Almost Tempered Distortion, defines a thermodynamic pressure $P(s)$ and finite-alphabet approximations, and constructs conformal/$s$-measures to connect the dimension to a critical exponent. The authors provide a complete upper and lower bound argument for product-of-digits sets, extending results beyond two digits and removing the need for Bounded Distortion Property. The approach combines a multi-dictionary covering argument for the upper bound with a Cantor-type construction and mass distribution for the lower bound, leveraging conformal measures and pressure continuity in non-BDP settings.

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 Lüroth map. In our approach we do not need to assume the Bounded Distortion Property (BDP).

Theorems & Definitions (16)

• Theorem 1.1
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5: Properties of Pressure Function