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

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 Schrödinger and the quasi-periodic wave operators. They have applications to the nonlinear quasi-periodic Schrödinger 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.