Uniformly discrete sequences in regions with tangential approach to the unit circle

A known result of Newman and Tse asserts that every uniformly discrete sequence contained in a Stolz angle is uniformly separated (see Newman, D.J., 1959, Interpolation in . Transactions of the American Mathematical Society, 92(3), 501–507; Tse, K.-F., 1971, Nontangential interpolating sequences and interpolation by normal functions. Proceedings of the American Mathematical Society, 29, 351–354). We prove that this statement no longer holds if the sequence is located in a tangential region of certain kind. It is well known that a uniformly discrete sequence need not be a Blaschke sequence. We show, however, that every uniformly discrete sequence inside a disc tangential to the unit circle must be a Blaschke sequence.