Symplectic non-squeezing for the cubic nonlinear Klein-Gordon equation on T3

We consider the periodic defocusing cubic nonlinear Klein-Gordon equation in three dimensions in the symplectic phase space H12(T3)×H−12(T3). This space is at the critical regularity for this equation, and in this setting there is no global well-posedness nor any uniform control on the local time of existence for arbitrary initial data. We prove a local-in-time non-squeezing result and a conditional global-in-time result which states that uniform bounds on the Strichartz norms of solutions imply global-in-time non-squeezing. As an application of the conditional result, we conclude non-squeezing for certain subsets of the phase space. The proofs rely on several approximation results for the flow, which we obtain using a combination of probabilistic and deterministic techniques.