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We study the problem of geometric zero packing, recently introduced byHedenmalm. There are two natural densities associated to this problem: thediscrepancy density $\rho_{\mathbb{H}}$, given by $$ \rho_{\mathbb{H}} =\liminf_{r\to 1^-} \inf_{f} \frac{\int_{\mathbb{D}(0,r)} \left((1-\lvertz\rvert^2) \lvert f(z)\rvert-1\right)^2 \frac{dA(z)}{1-\lvert z\rvert^2}}{\int_{\mathbb{D}(0,r)} \frac{dA(z)}{1-\lvert z\rvert^2}} $$ which measures thediscrepancy in optimal approximation of $(1-\lvert z\rvert^2)^{-1}$ with themodulus 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 deepconnections to the boundary behaviour of conformal mappings with$k$-quasiconformal extensions, which can be seen from the Hedenmalm's resultthat the universal asymptotic variance $\Sigma^2$ is related to$\rho_{\mathbb{H}}^*$ by $\Sigma^2=1-\rho_{\mathbb{H}}^*$. Here we prove thatin fact $\rho_{\mathbb{H}}=\rho_{\mathbb{H}}^*$, resolving a conjecture byHedenmalm in the positive. The natural planar analogues $\rho_{\mathbb{C}}$ and$\rho_{\mathbb{C}}^*$ to these densities make contact with work of Abrikosov onBose-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 withpolynomial growth control. As a consequence we obtain sufficiency results onthe degrees of approximately optimal polynomials.

Discrepancy densities for planar and hyperbolic Zero Packing