Discrepancy densities for planar and hyperbolic Zero Packing

We study the problem of geometric zero packing, recently introduced by Hedenmalm. There are two natural densities associated to this problem: the discrepancy density $\rho_{\mathbb{H}}$, given by $$ \rho_{\mathbb{H}} = \liminf_{r\to 1^-} \inf_{f} \frac{\int_{\mathbb{D}(0,r)} \left((1-\lvert z\rvert^2) \lvert f(z)\rvert-1\right)^2 \frac{dA(z)}{1-\lvert z\rvert^2}} {\int_{\mathbb{D}(0,r)} \frac{dA(z)}{1-\lvert z\rvert^2}} $$ which measures the discrepancy in optimal approximation of $(1-\lvert z\rvert^2)^{-1}$ with the modulus of polynomials $f$, and it's relative, the tight discrepancy density $\rho_{\mathbb{H}}^*$, which will trivially satisfy $\rho_{\mathbb{H}}\leq\rho_{\mathbb{H}}^*$. These densities have deep connections to the boundary behaviour of conformal mappings with $k$-quasiconformal extensions, which can be seen from the Hedenmalm's result that the universal asymptotic variance $\Sigma^2$ is related to $\rho_{\mathbb{H}}^*$ by $\Sigma^2=1-\rho_{\mathbb{H}}^*$. Here we prove that in fact $\rho_{\mathbb{H}}=\rho_{\mathbb{H}}^*$, resolving a conjecture by Hedenmalm in the positive. The natural planar analogues $\rho_{\mathbb{C}}$ and $\rho_{\mathbb{C}}^*$ to these densities make contact with work of Abrikosov on Bose-Einstein condensates. As a second result we prove that also $\rho_{\mathbb{C}}=\rho_{\mathbb{C}}^*$. The methods are based on Ameur, Hedenmalm and Makarov's H\"ormander-type $\bar\partial$-estimates with polynomial growth control. As a consequence we obtain sufficiency results on the degrees of approximately optimal polynomials.