Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces

Abstract

We give a new estimate for the ratio of s-dimensional Hausdorff measure \({\mathcal{H}^s}\) and (radius-based) packing measure \({\mathcal{P}^s}\) of a set in any metric space. This estimate is$$ \inf_{E}\frac{\mathcal{P}^s(E)}{\mathcal{H}^s(E)}\ge 1+\left(2-\frac{3}{2^{1/s}}\right)^s, $$where 0 < s < 1/2 and the infimum is taken over all metric spaces X and sets \({E\subset X}\) with \({0 < \mathcal{H}^s(E) < \infty}\). As an immediate consequence we improve the upper bound for the lower s-density of such sets in \({\mathbb{R}^n}\).