Necessary conditions on the support of RP-measures

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.