Annular functions in probability

Abstract

A function f f holomorphic in the open unit disk U U is said to be strongly annular if there exists a sequence { C n } \{ {C_n}\} of concentric circles converging outward to the boundary of U U such that the minimum of | f | |f| on C n {C_n} tends to infinity as n n increases. We show here that such functions with Maclaurin coefficients ± 1 \pm 1 form a residual set in the space of functions with coefficients ± 1 \pm 1 . We also show that the set of t t in [ 0 , 1 ] [0,1] for which ∑ r n ( t ) z n \sum {{r_n}(t){z^n}} is strongly annular ( r n {r_n} is the n n th Rademacher function) is residual, and measurable with measure either 0 0 or 1 1 .