  We consider the pointwise convergence problem for the solution of
Schr\"odinger-type equations along directions determined by a given compact
subset of the real line. This problem contains Carleson's problem as the most
simple case and was studied in general by Cho--Lee--Vargas. We extend their
result from the case of the classical Schr\"odinger equation to a class of
equations which includes the fractional Schr\"odinger equations. To achieve
this, we significantly simplify their proof by completely avoiding a time
localization argument.
