Catching sets with quasicircles (Q1125257)

The following two theorems are proved: Theorem 1. For a subset \(E\) of \(\mathbb{C}\) the following statements are equivalent: i) \(E\) has empty interior and uniform complement. ii) \(E\) is uniformly disconnected. iii) \(E\) is quasiconformally equivalent to a porous subset of \(R\). The various constants depend only on each other. Theorem 2. For a compact set \(K\) in \(\mathbb{C}\) whose interior is empty the following statements are equivalent: i) \(K\) is uniformly perfect and has uniform complement. ii) \(K\) is both uniformly perfect and uniformly disconneted. iii) \(K\) is quasiconformally equivalent to the usual Cantor middle-third set. The various constants depend only on each other.