Dichotomy laws for the Hausdorff measure of shrinking target sets in $\beta$-dynamical systems

TL;DR

This paper establishes a dichotomy law for the Hausdorff measure of shrinking target sets in $eta$-dynamical systems, revealing that the measure is either zero or full based on series convergence, thus advancing measure theory in this context.

Contribution

It provides the first complete measure theoretic characterization of shrinking target sets in $eta$-dynamical systems, linking Hausdorff measure to series convergence.

Findings

Hausdorff measure is zero or full depending on series divergence or convergence.

Provides a complete measure theoretic description for these sets.

Extends understanding of measure theory in $eta$-dynamical systems.

Abstract

In this paper, we investigate the Hausdorff measure of shrinking target sets in $\beta$-dynamical systems. These sets are dynamically defined in analogy to the classical theory of weighted and multiplicative approximation. While the Lebesgue measure and Hausdorff dimension theories for these sets are well-understood, the Hausdorff measure theory in even one-dimensional settings remains unknown. We show that the Hausdorff measure of these sets is either zero or full depending upon the convergence or divergence of a certain series, thus providing a rather complete measure theoretic description of these sets.