Rational approximation on the unit sphere in \(\mathbb{C}^2\) (Q1885567)

Let \(X\) be a compact subset of \(\mathbb C^n\) and let \(R(X)\) be the closure in \(C(X)\) of the set of rational functions holomorphic in a neighborhood of \(X\). The authors are interested in finding conditions on \(X\) which imply \(R(X)=C(X)\). Let \(\hat{X_r}=\{z\in\mathbb C^n:\;\text{every polynomial}\;Q\;\text{with}\;Q(z)=0\;\text{vanishes at some point of}\;X\}\). The condition \(X=\hat{X_r}\) (i.e. that \(X\) is rationally convex) is necessary for \(R(X)=C(X)\). If \(n=1\), every compact set \(X\) is rationally convex. However, there are known examples of sets without interior for which \(R(X)\neq C(X)\) (e.g. the Alice Roth's Swiss cheese). On the other hand, in virtue of the Hartogs-Rosenthal theorem, if the two-dimensional Lebesgue measure of \(X\) is zero, then \(R(X)=C(X)\). \textit{R. F. Basener} [Trans. Am. Math. Soc. 182, 353--381 (1973; Zbl 0239.46051)] has given examples of rationally convex sets \(X\subset\mathbb C^2\) of the form \(\{(z,w)\in\partial B:\;z\in E\}\), where \(E\subset\mathbb C\) is a suitable Swiss cheese, such that \(R(X)\neq C(X)\). These sets have the property that \(\sigma(X)>0\), where \(\sigma\) is three-dimensional Hausdorff measure on \(\partial B\). In view of the one-dimensional Hartogs-Rosenthal theorem, it is reasonable to conjecture that if \(X=\hat{X_r}\) and \(\sigma(X)=0\), then \(R(X)=C(X)\). In the paper, the authors give several contributions to the study of this question.