Motivated by work of Cantat-Dujardin, we study orbits by translations in K3 surfaces with two elliptic fibrations. We prove in particular that all orbits are infinite away from a proper Zariski-closed subset.Among the tools, beyond the Pila-Wilkie counting, are a new version of Ax-Schanuel by Bakker-Tsimerman and a(n old) theorem of Shioda on ramification of sections of the Legendre family.This is joint work with Corvaja and Tsimerman.

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