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

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.