Cauchy transforms of characteristic functions and algebras generated by inner functions

We prove that Cauchy transforms of characteristic functions of subsets of positive measure of the unit circle are equidistributed in the unit disk in the sense that the -closure of the polynomial algebra in these Cauchy transforms coincides with the -closure of the polynomial algebra in a canonical inner function. As a corollary to this result we find conditions describing when the polynomial algebra in two singular inner functions determined by point masses is dense in the Hardy spaces .