Lower bounds on concentration through Borel transforms and quantitative singularity of spectral measures near the arithmetic transition

TL;DR

This paper introduces new analytical tools to study the spectral measures of certain operators near the arithmetic transition, providing bounds on their dimensions and partial localization results.

Contribution

It develops a general criterion for lower bounds on Borel measure concentrations using Borel transform boundary behavior, and applies it to spectral measures in the hyperbolic regime.

Findings

First bounds on packing and multifractal dimensions near the transition

Partial localization of generalized eigenfunctions in the singular continuous regime

A new criterion for measure concentration bounds based on Borel transforms

Abstract

We develop tools to study arithmetically induced singular continuous spectrum in the neighborhood of the arithmetic transition in the hyperbolic regime. This leads to first transition-capturing upper bounds on packing and multifractal dimensions of spectral measures. We achieve it through the proof of partial localization of generalized eigenfunctions, another first result of its kind in the singular continuous regime. The proof is based also on a general criterion for lower bounds on concentrations of Borel measures as a corollary of boundary behavior of their Borel-type transforms, that may be of wider use and independent interest.