Diophantine estimates on shifts of trigonometric polynomials on $\mathbb{T}^d$

TL;DR

This paper develops Diophantine estimates for shifts of trigonometric polynomials on multi-dimensional tori, with applications to spectral analysis and nonlinear quasi-periodic Schrödinger and wave equations.

Contribution

It extends Diophantine estimates to the nonlinear setting, enabling analysis of nonlinear quasi-periodic PDEs using spectral methods.

Findings

Establishes Diophantine estimates for shifts of trigonometric polynomials on  torus.

Applies these estimates to spectral analysis of quasi-periodic Schrdinger and wave operators.

Enables extension of Bourgain's results to nonlinear quasi-periodic PDEs.

Abstract

We establish Diophantine type estimates on shifts of trigonometric polynomials on the torus $\mathbb{T}^d$, as well as that of their square roots. These estimates arise from the spectral analysis of the quasi-periodic Schr\"odinger and the quasi-periodic wave operators. They have applications to the nonlinear quasi-periodic Schr\"odinger equations (NLS) and the nonlinear quasi-periodic wave equations (NLW). One could now, for example, extend the result of Bourgain (Geom. Funct. Anal. 17(3): 682-706, 2007) to the nonlinear setting.