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

Svetlana Jitomirskaya, Wencai Liu, Serguei Tcheremchantsev

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.

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

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.
Paper Structure (12 sections, 24 theorems, 169 equations)

This paper contains 12 sections, 24 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fractal dimensions of measures and proof of corollaries
  4. $m$-functions and power-law subordinacy
  5. Lyapunov exponent
  6. Proof of Theorem \ref{['D-']}
  7. Proof of Theorem \ref{['D+']}
  8. Proof of Theorem \ref{['upD+']}
  9. Basics for applications to the almost Mathieu operator
  10. Partial localization
  11. Estimates on all solutions
  12. Proof of Theorems \ref{['Maintheoremp']} and \ref{['Maintheoremp1']}

Key Result

Theorem 1.1

For $\alpha\in \mathbb{R}\backslash \mathbb{Q}$, the following holds for the almost Mathieu operator Def.AMO, any $\alpha$-Diophantine phase $\theta$ and any $\varphi\in \ell^2(\mathbb{Z})$,

Theorems & Definitions (43)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • ...and 33 more