Annular functions in probability

Abstract:

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