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

Runge Approximation and Stability Improvement for a Partial Data Calder\'on Problem for the Acoustic Helmholtz Equation