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Two results concerning the boundary zero sets of analytic functions on the unit disk      Δ  \Delta       are proved. First we consider nonconstant analytic functions      f f       on      Δ  \Delta       for which the radial limit function      f ∗  {f^{\ast }}       is defined at each point of the unit circumference      C C      . We show that a subset      E E       of      C C       is the zero set of      f ∗  {f^{\ast }}       for some such function      f f       if and only if it is a      G δ  {\mathcal {G}_\delta }       that is not metrically dense in any open arc of      C C      . We then give a precise version of an asymptotic radial uniqueness theorem and its converse. The constructions given in the proofs of each of these theorems employ an approximation theorem of Arakeljan.

Some results concerning the boundary zero sets of general analytic functions