Davis–Lelièvre conjecture on asymmetric combinatorial periods in the regular pentagon

Prove that every positive even integer except 2, 12, 14, and 18 occurs as the combinatorial period length of an asymmetric closed billiard trajectory on the regular pentagon, where the combinatorial period length is defined as the number of wall collisions before first return and asymmetry means the trajectory lacks fivefold rotational symmetry.

Background

Periodic billiard trajectories in the regular pentagon can be studied via saddle connections on the associated golden L translation surface. Davis and Lelièvre conducted extensive numerical experiments and proposed a precise characterization for which even integers appear as combinatorial period lengths of asymmetric closed trajectories on the pentagon.

The present paper proves a density-one version of this conjecture, showing that almost every even integer arises, but it does not establish the full statement; the detailed local-global framework and strong approximation analysis indicate that the only local obstruction is parity.