The property of unique continuation in certain spaces spanned by rational functions on compact nowhere dense sets

It has been known for over a century that certain large classes of functions defined on a compact nowhere dense subset $X$ of the complex plane, and obtained as limits of analytic functions in various metrics, can sometimes inherit the property of unique continuation characteristic of the approximating family. The first example of the transfer of the uniqueness property in this way to $R(X)$, the space of functions that can be uniformly approximated on $X$ by a sequence of rational functions whose poles lie outside of $X$, was obtained by M. V. Keldysh around 1940, but apparently never published. Years later in 1975 A. A. Gonchar exhibited a qualitatively definitive improvement of Keldysh's example, and our goal here is to extend that result to $R^p(X,dA), p\geq 2,$ the evidently larger space obtained as the closure of the rational functions in $L^p(X,dA)$, where $dA$ denotes 2-dimensional Lebesgue, or area, measure.