Bourgain's counterexample in the sequential convergence problem for the Schrödinger equation

We study the problem of pointwise convegence for the Schrödinger operator on ${\mathbb{R}}^n$ along time sequences. We show that the sharp counterexample to the sequential Schrödinger maximal estimate given recently by Li, Wang and Yan based in the construction by Lucà and Rogers can also be achieved with the construction of Bourgain, and we extend it to the fractal setting.