Decoupling and Discrete Strichartz Estimates for Dispersive Equations on the Torus

Abstract

This paper proves weighted discrete Strichartz, or equivalently discrete restriction, estimates for two-dimensional exponential sums with fractional and perturbed polynomial phases. The new results cover three related classes of curves: the fractional Schr\"odinger-type curve $(n,n^{1+\nu})$, the fully fractional curve $(n^{1+\mu},n^{1+\nu})$ with $0<\mu<\nu$, and bounded perturbations of the polynomial curve $(n,n^k)$. In each case we obtain $\ell^2$-weighted $L^p$ estimates with decoupling-type dependence on the frequency scale, including both low- and high-$p$ regimes. The proofs are based on finite-type decoupling, together with localization and rescaling arguments adapted to curvature degeneracy, anisotropic homogeneity, and bounded perturbations of the phase. The introduction frames these estimates from the viewpoint of decoupling theory, with arithmetic results obtained by efficient congruencing used only as comparison points for the known cases.