  We study the Debye mass, $m_D$, and the topological susceptibility, $\chi$,
at high temperatures in non-abelian gauge theory. Both exhibit, at some order
in the perturbation expansion, infrared sensitivity. As a result, a
perturbative analysis can at best provide an estimate of these quantities,
subject to some uncertainty. The size of these uncertainties, particularly in
the case of $\chi$, has been the subject of some debate. For the perturbative
free energy, reframing an analysis of Braaten and Pisarski, the estimate and
the associated error, can be understood in terms of a {\it Wilsonian} effective
action for the low energy effective three dimensional theory. This action can
be obtained completely from a perturbative calculation, which terminates at a
finite order. This action provides the desired estimate. The size of the error
follows from dimensional analysis in the low energy theory. The Debye length
computation and its error can be obtained from a similar study of a
non-relativistic effective theory for an adjoint scalar in three dimensions.
$\chi$ requires a four dimensional analysis involving finite temperature
instantons, but again the dominant sources of uncertainty are three
dimensional, and we provide a procedure to estimate and an associated error.
This uncertainty is of order $g^4 $ relative to the leading semiclassical
result, and in situations of interest is small.
