  In this article, we discuss quantitative Runge approximation properties for
the acoustic Helmholtz equation and prove stability improvement results in the
high frequency limit for an associated partial data inverse problem modelled on
\cite{AU04, KU19}. The results rely on quantitative unique continuation
estimates in suitable function spaces with explicit frequency dependence. We
contrast the frequency dependence of interior Runge approximation results from
non-convex and convex sets.
