Symplectic non-squeezing for the cubic nonlinear Klein-Gordon equation on $\mathbb{T}^3$

We consider the periodic defocusing cubic nonlinear Klein-Gordon equation in three dimensions in the symplectic phase space $H^{\frac{1}{2}}(\mathbb{T}^3) \times H^{-\frac{1}{2}}(\mathbb{T}^3)$. 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 a consequence of the conditional result, we conclude non-squeezing for certain subsets of the phase space, and in particular, we prove small data non-squeezing for long times. The proofs rely on several approximation results for the flow, which we obtain using a combination of probabilistic and deterministic techniques.