An Upbound of Hausdorffs Dimension of the Divergence Set of the fractional Schrodinger Operator on $H^s(mathbb R^n)

This paper shows $$ sup_{fin H^s(mathbb{R}^n)}dim _Hleft{xinmathbb{R}^n: lim_{trightarrow0}e^{it(-Delta)^alpha}f(x) eq f(x)right}leq n+1-frac{2(n+1)s}{n} text{under} begin{cases} ngeq2; alpha>frac12; frac{n}{2(n+1)}<sleqfrac{n}{2} . end{cases} $$