  The purpose of this note is to provide a summary of the recent work of the
authors on two variations of the pointwise convergence problem for the
solutions to the fractional Schr\"odinger equations; convergence along a
tangential line and along a set of lines, as exhibiting some new results in
each setting. For the former case, we make a simple observation on a path along
a tangential curve of exponential order. We discuss counterexamples for the
latter case that show some of the known smooth regularities are essentially
optimal.
