Bloch functions, asymptotic variance, and geometric zero packing

We study a new type of extremal problem in complex analysis, referred to as "geometric zero packing", which is the hyperbolic analogue of a problem considered by Abrikosov in the 1950s concerning Bose-Einstein condensates. We relate the corresponding minimal discrepancy density with the asymptotic variance for Bloch functions of the form "Bergman projection of bounded functions" and obtain a corresponding identity. Together with related work of Ivrii, this gives the asymptotic behavior of the universal quasiconformal integral means spectrum for small values of quasiconformality k and small exponents t. In particular, the conjectured behavior is shown to be smaller than conjectured by Prause and Smirnov, which also shows that there are no quasidisks with dimension 1+k^2, at least for small k.

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