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We obtain partial improvement toward the pointwise convergence problem ofSchr\"odinger solutions, in the general setting of fractal measure. Inparticular, we show that, for $n\geq 3$, $\lim_{t \to 0} e^{it\Delta}f(x) =f(x)$ almost everywhere with respect to Lebesgue measure for all $f \in H^s(\mathbb{R}^n)$ provided that $s>(n+1)/2(n+2)$. The proof uses linear refinedStrichartz estimates. We also prove a multilinear refined Strichartz usingdecoupling and multilinear Kakeya.

Pointwise convergence of Schr\"odinger solutions and multilinear refined Strichartz estimates