A Comparison of Gauge Dimension and Effective Dimension

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