Dimension of divergence sets of oscillatory integrals with concave phase

Abstract: We study the Hausdorff dimension of the sets on which the pointwise convergence of the solutions to the fractional Schr\"odinger equation $e^{{it(-\Delta)^\frac} m2}f$ fails when $m\in(0,1)$ in one spatial dimension. The pointwise convergence along a non-tangential curve and a set of lines are also considered, where we find different nature from the case when $m\in(1,\infty)$.