= Pressure upper bound for Hausdorff dimension
{title2=$P(\gamma_*)=0\ \Longrightarrow\ \dim_H F\leq\gamma_*$}
Suppose shrinking interval covers of $F$ have length-power sums with logarithmic growth rate $P(\gamma)$, and $P$ is strictly decreasing with zero $\gamma_*$. For every $\gamma>\gamma_*$ those sums tend to zero exponentially. The definition of <Hausdorff measure> then gives $\mathcal H^\gamma(F)=0$, hence the displayed upper bound. A matching lower bound requires additional information and does not follow from this covering argument alone.
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