A note on $L^p$-bounded point evaluations for polynomials

Abstract:

We construct a compact nowhere dense subset $K$ of the closed unit disk $\bar {\mathbb D}$ in the complex plane $\mathbb C$ such that $R(K) = C(K)$ and bounded point evaluations for $P^t(dA | _K), ~ 1 \le t < \infty ,$ is the open unit disk $\mathbb D.$ In fact, there exists $C=C(t) > 0$ such that \[ \ \int _{\mathbb D} |p|^t dA \le C \int _K |p|^t dA, \] for $1 \le t < \infty$ and all polynomials $p.$