Abstract
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 characterautomorphic 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 CE. 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.
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