Talbot effect on the sphere and torus for d≥2

We utilize exponential sum techniques to obtain upper and lower bounds for the fractal dimension of the graph of solutions to the linear Schrödinger equation on S^d and T^d. Specifically for S^d, we provide dimension bounds using both L^p estimates of Littlewood-Paley blocks, as well as assumptions on the Fourier coefficients. In the appendix, we present a slight improvement to the bilinear Strichartz estimate on S² for functions supported on the zonal harmonics. We apply this to demonstrate an improved local well-posedness result for the zonal cubic NLS when d=2, and a nonlinear smoothing estimate when d≥2. As a corollary of the nonlinear smoothing for solutions to the zonal cubic NLS, we find dimension bounds generalizing the results of Erdoğan and Tzirakis (Math Res Lett 20(6): 1081–1090, 2013) for solutions to the cubic NLS on T. Additionally, we obtain several results on T^d generalizing the results of the d=1 case.