Bloch functions, asymptotic variance, and geometric zero packing

TL;DR

This paper introduces a new extremal problem called geometric zero packing in complex analysis, relating it to Bloch functions and quasiconformal spectra, with implications for the geometry of quasidisks.

Contribution

It establishes a novel connection between geometric zero packing, asymptotic variance of Bloch functions, and quasiconformal spectra, providing new insights and bounds.

Findings

Minimal discrepancy density relates to asymptotic variance of Bloch functions.

The conjectured quasiconformal spectrum behavior is smaller than previously thought.

No quasidisks with dimension 1+k^2 exist for small k.

Abstract

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.