Effective Dimension s and Gauge Functions

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.

Figures (1)

• Figure 1: Tree $T$

Theorems & Definitions (23)

• Definition 1.1
• Definition 1.2
• Definition 1.3: beresnevich2006measure
• Theorem 2.1: Cai, Hartmaniscai1994hausdorff; Reimannreimann2004computability
• Theorem 2.2
• proof
• Claim
• proof : Proof of the claim
• Claim
• proof : Proof of the Claim.