Non-resonant conditions for the Klein-Gordon equation on the circle

We consider the infinite dimensional vector of frequencies $\omega(m)=( \sqrt{j^2+m})_{j\in \mathbb{Z}}$, $m\in [1,2]$ arising form a linear Klein-Gordon equation on the one dimensional torus and prove that there exists a positive measure set of masses $m'$s for which $\omega(m)$ satisfies a diophantine condition similar to the one introduced by Bourgain in (JFA, 2005), in the context of Schr\"odinger equation with convolution potential. The main difficulties we have to deal with are the asymptotically linear nature of the (infinitely many) $\omega_{j}'$s and the degeneracy coming from having only one parameter at disposal for their modulation. As an application we provide estimates on the inverse of the adjoint action of the associated quadratic Hamiltonian on homogenenous polynomials of any degree in Gevrey category.