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 S 2 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.