Diophantine frequency vector ($|k\cdot \omega |\ge \gamma |k{|}_1^{-\tau }$)

A vector $\omega \in {\mathbb{R}}^n$ satisfying $|k\cdot \omega |\ge \gamma |k{|}_1^{-\tau }$ for all nonzero integer vectors $k$, with $\gamma >0$. This excludes resonances and controls the small divisors in analytic cohomological equations on a Diophantine torus. Different choices of norm change the constant but not the nature of the condition.

A denominator such as $k\cdot \omega$ which may become small for large integer vectors $k$. Inverting a directional derivative on a torus divides the nonzero Fourier coefficients by $ik\cdot \omega$. A Diophantine condition bounds this loss polynomially, so exponential Fourier decay still gives convergence on a smaller complex strip.