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

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