For α-Hölder curves γ(t)=(t^{α_1},...,t^{α_n}) with α=min α_j <1/2, the critical Sobolev regularity for a.e. convergence of Schrödinger solutions is s=max{(1-2α)/2, n/(2(n+1))}.
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On the Pointwise Convergence of Solutions to the Schr\"odinger Equation Along Certain Highly Tangential Curves
For α-Hölder curves γ(t)=(t^{α_1},...,t^{α_n}) with α=min α_j <1/2, the critical Sobolev regularity for a.e. convergence of Schrödinger solutions is s=max{(1-2α)/2, n/(2(n+1))}.