We consider the pointwise convergence problem for the solution of Schrödinger-type equations along directions determined by a given compact subset of the real line. This problem contains Carleson's problem, which has been recently very actively studied and has progressed a lot, as the most simple case and was studied in general by Cho--Lee--Vargas. I will talk about my recent work that extends their result from the case of the classical Schrödinger equation to a class of equations which includes the fractional Schrödinger equations.