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 $m\in(1,\infty)$.