Talbot Effect on the Sphere and Torus for $d\geq 2$

Abstract

We utilize exponential sum techniques to obtain upper and lower bounds for the fractal dimension of the graph of solutions to the linear Schr\"odinger equation on $\mathbb{S}^d$ and $\mathbb{T}^d$. Specifically for $\mathbb S^d$, 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 $\mathbb{S}^2$ 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 $d\geq 2$. As a corollary of the nonlinear smoothing for solutions to the zonal cubic NLS, we find dimension bounds generalizing the results of the first author and Tzirakis for solutions to the cubic NLS on $\mathbb{T}$. Additionally, we obtain several results on $\mathbb{T}^d$ generalizing the results of the $d=1$ case.