A Comparison of Gauge Dimension and Effective Dimension

TL;DR

This paper investigates the properties of sets of real numbers classified by their effective dimension and their relationship with well-approximable reals, using Hausdorff measure to distinguish between these classes.

Contribution

It characterizes the gauge profiles of sets of reals with specific effective dimensions and establishes a separation between these sets and well-approximable reals via Hausdorff measure.

Findings

Separation between $ ext{D}_{ ext{leq} s}$ and $W(2/s)$ in Hausdorff measure.

Characterization of the gauge profile of $ ext{D}_s$ and $ ext{D}_{ ext{leq} s}$.

Insights into the structure of reals with given effective dimensions.

Abstract

We characterize the gauge profile of $\mathcal{D}_s$, the set of reals with effective dimension $s$, and $\mathcal{D}_{\leq s}$, the set of reals with effective dimension $\leq s$. Let $W(s)$ be the set of reals that are $s$-well approximable. This gives us a separation between $\mathcal{D}_{\leq s}$ and $W(2/s)$ in terms of Hausdorff measure.