Effective Mordell–Lang for the square of an elliptic curve

Develop an effective Mordell–Lang theorem for rational points on the square E^2 of an elliptic curve, sufficient to yield an analogue of the effective height lower-bound theorem proved for finitely generated subgroups of G_m^2.

Background

Appendix 1 proves an effective height lower bound for rational functions on finitely generated subgroups of the two-dimensional torus G_m^2, using effective results for points on certain subvarieties of tori.

The authors note that the proof does not extend to the square of an elliptic curve because the corresponding effective Mordell–Lang theory is unavailable. An effective result in that setting would provide an elliptic analogue of the toric theorem.