In this paper, the authors study the almost everywhere pointwise convergence problem along a class of restricted curves in $ℝ\times ℝ$ given by {(y, t): y∈Γ(x, t)} for each t∈[0, 1], where Γ(x, t)={γ(x, t, θ): θ∈Θ} for a given compact set Θ in $ℝ$ of the fractional Schrödinger propagator and Boussinesq operator. They focus on the relationship between the upper Minkowski dimension of Θ and the optimal s for which $\underset{\begin{array}{l}y\in \Gamma (x,t)\\ (y,t)\to (x,0)\end{array}}{\lim }{e}^{it{(\sqrt{-\Delta })}^a}f(y)=f(x),\underset{\begin{array}{l}y\in \Gamma (x,t)\\ (y,t)\to (x,0)\end{array}}{\lim }{\mathcal{B}}_{t}f(y)=f(x),a.e.,$ whenever $ｆ\in {Ｈ}^{ｓ}（ℝ）$.