Boundary uniqueness theorems for meromorphic functions

A study is made of sets of uniqueness for the class of arbitrary meromorphic functions on the disk and for the limit values over $h$-angles (domains with zero angle on the boundary and with form determined by a function $h(x)$). The sets of uniqueness are characterized with the help of the concepts of $h$-indecomposability of $h$-regularity, introduced and studied in this article. These concepts turn out to be intermediate between measure and category. The concept of the porosity of a set served as a starting point for the definition of the property of $h$-indecomposability. The central result in this paper is the following:
Theorem. Let $\mathscr F$ be the class of all meromorphic functions $f(z)$ on the unit disk. A set $E$ on the boundary of the disk is a set of uniqueness for the class $\mathscr F$ and for the limit values over $h$-angles if and only if $E$ is $h$-indecomposable.
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