  We give necessary conditions for when a subset of $\mathbb{T}^n$ can contain
the support of some non-zero RP-measure. Among other things we show that the
support of a positive RP-measure cannot be contained in reflections of inverse
images of half-planes by certain functions in $A(\mathbb{D}^n)$, sets of linear
measure zero, and when $n=2$, graphs of strictly increasing functions. For
$n=2$ we also prove that failure to contain the support of any positive
RP-measure implies that restrictions of functions in a subspace of
$A(\mathbb{D}^2)$ are uniformly dense among the continuous functions on a
related set.
