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

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 $P^t(dA \vert _K), ~ 1 \le t < \infty ,$ is the open unit disk $\mathbb{D}.$ In fact, there exists $C=C(t) > 0$ such that

$$\ \int _{\mathbb{D}} \vert p\vert^t dA \le C \int _K \vert p\vert^t dA,$$

for $1 \le t < \infty$ and all polynomials $p.$