On the Direct Cauchy Theorem in Widom Domains: Positive and Negative Examples

We discuss several questions which remained open in our joint work with M. Sodin "Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character--automorphic functions". In particular, we show that there exists a non-homogeneous set $E$ such that the Direct Cauchy Theorem (DCT) holds in the Widom domain $\bbC\setminus E$. On the other hand we demonstrate that the weak homogeneity condition on $E$ (introduced recently by Poltoratski and Remling) does not ensure that DCT holds in the corresponding Widom domain.