An explicit exact Hausdorff gauge for SLE

For each fixed $0<\kappa <8$, let $d=1+\kappa /8$. The gauge $h(r)=r^d(\log \log (1/r){)}^{(2-d)/2}$ at sufficiently small radii almost surely gives positive finite Hausdorff measure to every nontrivial positive-time compact segment of chordal SLE${}_{\kappa }$. This gives an explicit solution to Schramm's Hausdorff-measure problem in this parameter range. The entire trace has finite expected measure in every bounded disk. The exponent-one iterated-logarithm gauge suggested by Schramm is not sigma-finite on any such segment.