On interpolation free disks of polynomials converging maximally to power series

We construct a power series $f$ with radius $R$ of convergence, $0 \lt R \lt \infty $, such that for any $\sigma $, $0 \lt \sigma \lt R$, there exists a subset $\Lambda \subset \mathbb N$, a parameter $r_\sigma $, $0 \lt r_\sigma \lt \sigma $, and a seque