On Bourgain's counterexample for the Schr\"odinger maximal function

TL;DR

This paper rigorously constructs and optimizes Bourgain's counterexample to demonstrate limitations in pointwise convergence of Schr"odinger solutions for initial data in Sobolev spaces, combining analysis and number theory techniques.

Contribution

It provides a detailed, first-principles derivation and optimization of Bourgain's counterexample, clarifying its construction and implications.

Findings

Counterexample confirms limitations in pointwise convergence

Method combines analysis and number theory techniques

Optimized construction enhances understanding of Schr"odinger convergence issues

Abstract

This paper provides a rigorous derivation of a counterexample of Bourgain, related to a well-known question of pointwise a.e. convergence for the solution of the linear Schr\"odinger equation, for initial data in a Sobolev space. This counterexample combines ideas from analysis and number theory, and the present paper demonstrates how to build such counterexamples from first principles, and then optimize them.