Generic zero-Hausdorff and one-packing spectral measures

TL;DR

This paper demonstrates that in certain metric spaces of self-adjoint operators, the spectral measures with both zero upper-Hausdorff and one lower-packing dimensions form a dense $G_\delta$ set, with applications to limit-periodic operators.

Contribution

It establishes the genericity of spectral measures with specific fractal dimensions in metric spaces of operators, extending understanding of spectral measure properties.

Findings

Spectral measures with zero upper-Hausdorff and one lower-packing dimensions are dense in certain operator spaces.

The set of such operators contains a dense $G_\delta$ subset.

Applications include properties of limit-periodic operators.

Abstract

For some metric spaces of self-adjoint operators, it is shown that the set of operators whose spectral measures have simultaneously zero upper-Hausdorff and one lower-packing dimensions contains a dense $G_\delta$ subset. Applications include sets of limit-periodic operators.