  We study the almost periods of the eigenmodes of flat planar manifolds in the
high energy limit. We prove in particular that the Gaussian Arithmetic Random
Waves replicate almost identically at a scale at most ${\ell}$n := n -- 1 2 exp
(Nn), where Nn is the number of ways n can be written as a sum of two squares.
It provides a qualitative interpretation of the full correlation phenomenon of
the nodal length, which is known to happen at scales larger than ${\ell}$ ' n
:= n --1/2 N A n. We provide also a heuristic with a toy model pleading that
the minimal scale of replication should be closer to ${\ell}$ ' n than
${\ell}$n. Contents 1. Introduction 1 2. Almost periodicity and replication 6
3. Dirichlet's theorem for almost periodic fields 13 4. Replication of the
nodal lines 15 5. Optimality of Dirichlet's approximation theorem 19 6.
Appendix 20 References 28
