  This note presents some of the results obtained in arXiv:2207.05410 and it
has beenthe object of a talk of the second author during the Journ\'ees
"\'Equations auxD\'eriv\'ees Partielles" (Obernai, june 2022). We study
properties of geodesics that are orthogonal to two convex subsets of the flat
torus. We discuss meromorphic properties of a geometric Epstein zeta function
associated to the set of lengths of such orthogeodesics. We also define the
associated length distribution and discuss singularities of its Fourier
transform. Our analysis relies on a fine study of the dynamical correlation
function of the geodesic flow on the torus and the definition of anisotropic
Sobolev spaces that are well-adapted to this integrable dynamics.
