Length spectrum of periodic rays for billard flow

We study for several compact strictly convex disjoint obstacles the length spectrum $\mathcal{L}$ formed by the lengths of all primitive periodic reflecting rays. We prove the existence of sequences $\{{\mathrm{\ell }}_j\},\{m_j\}$ with ${\mathrm{\ell }}_j\in \mathcal{L},m_j\in \mathbb{N}$ such that the condition (LB) related to the dynamical zeta function ${\eta }_D(s)$ is satisfied. This condition implies the existence of lower bounds for the number of the scattering resonances for Dirichlet Laplacian. We construct such sequences under some separation condition for a small subset of $\mathcal{L}$ corresponding to lengths of the periodic rays with even reflexions. Our separation condition is weaker than the assumption of exponentially separated length spectrum $\mathcal{L}\mathrm{.}$ Moreover, we show that the periodic orbits in the phase space are exponentially separated.