Discrepancy densities for planar and hyperbolic zero packing

We study the problem of geometric zero packing, recently introduced by Hedenmalm [7]. There are two natural densities associated with this problem: the discrepancy density ρH, given byρH=liminfr→1−inff⁡∫D(0,r)((1−|z|2)|f(z)|−1)2dA(z)1−|z|2∫D(0,r)dA(z)1−|z|2 which measures the discrepancy in optimal approximation of (1−|z|2)−1 with the modulus of polynomials f, and its relative, the tight discrepancy density ρH⁎, which will trivially satisfy ρH≤ρH⁎. These densities have deep connections to the boundary behaviour of conformal mappings with k-quasiconformal extensions, which can be seen from Hedenmalm's result that the universal asymptotic variance Σ2 is related to ρH⁎ by Σ2=1−ρH⁎. Here we prove that in fact ρH=ρH⁎, resolving a conjecture by Hedenmalm in the positive. The natural planar analogues ρC and ρC⁎ to these densities make contact with work of Abrikosov on Bose-Einstein condensates. As a second result we prove that also ρC=ρC⁎. The methods are based on Ameur, Hedenmalm and Makarov's Hörmander-type ∂¯-estimates with polynomial growth control [2]. As a consequence we obtain sufficiency results on the degrees of approximately optimal polynomials.