2016/05/27 by Wennman, Aron
#30C62 (Primary) #30C70 #30H20 (Secondary) #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1605.08674
We study the problem of geometric zero packing, recently introduced by Hedenmalm. There are two natural densities associated to this problem: the discrepancy density ρℍ, given by ρℍ = \liminfr→ 1- inff \frac∫_\mathbbD(0,r) ((1-| z|2) | f(z)|-1)2 (dA(z))/(1-| z|2) ∫_\mathbbD(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 it's 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 the 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. As a consequence we obtain sufficiency results on the degrees of approximately optimal polynomials.