Abstract In this paper, the authors study the almost everywhere pointwise convergence problem along a class of restricted curves in \(\mathbb{R} \times \mathbb{R}\) given by \(\{(y,t) : y \in \Gamma(x,t)\}\) for each \(t \in [0,1]\), where \(\Gamma(x,t) = \{\gamma(x,t,\theta) : \theta \in \Theta\}\) for a given compact set \(\Theta\) in \(\mathbb{R}\) of the fractional Schr?dinger propagator and Boussinesq operator. They focus on the relationship between the upper Minkowski dimension of \(\Theta\) and the optimal \(s\) for which

\[ \lim_{\substack{y \in \Gamma(x,t) \\ (y,t) \to (x,0)}} \mathrm{e}^{\mathrm{i}t (-\Delta)^{\alpha}} f(y) = f(x), \quad \lim_{\substack{y \in \Gamma(x,t) \\ (y,t) \to (x,0)}} \mathcal{B}_t f(y) = f(x), \quad \text{a.e.,} \] whenever \(f \in H^s(\mathbb{R})\).