Counterexamples for the fractal Schr\"odinger convergence problem with an intermediate space trick

TL;DR

This paper constructs counterexamples for the fractal Schrödinger convergence problem using advanced techniques, confirming the limits of regularity and explicitly analyzing divergence sets with Hausdorff dimension calculations.

Contribution

It introduces a novel combination of fractal extensions and intermediate space tricks to produce counterexamples, advancing understanding of convergence issues in fractal Schrödinger problems.

Findings

Counterexamples match the regularity of previous weighted L^2 restriction counterexamples.

Explicit construction and Hausdorff dimension computation of divergence sets.

Application of the Mass Transference Principle to analyze divergence sets.

Abstract

We construct counterexamples for the fractal Schr\"odinger convergence problem by combining a fractal extension of Bourgain's counterexample and the intermediate space trick of Du--Kim--Wang--Zhang. We confirm that the same regularity as Du's counterexamples for weighted $L^2$ restriction estimates is achieved for the convergence problem. To do so, we need to construct the set of divergence explicitly and compute its Hausdorff dimension, for which we use the Mass Transference Principle, a technique originated from Diophantine approximation.