Geometric lifting and Freiman's $3k-4$ theorem in compact connected abelian groups

We develop a geometric lifting method for inverse sumset problems in compact connected abelian groups. The first result is an analogue of Freiman’s $3k-4$ theorem, that is every compact set $A\subseteq G$ of sufficiently small Haar measure satisfying ${\mu }_G(A+A)<3{\mu }_G(A)$ is contained in a one dimensional Bohr set of measure at most ${\mu }_G(A+A)-{\mu }_G(A)$. This resolves a question of Christ and Iliopoulou. The proof combines Bilu’s theorem with a geometric refinement of the spillover argument. We also establish a sharp projection theorem. Under a continuous surjective homomorphism with connected kernel, a compact set of sufficiently small positive measure and doubling at most $K$, where $2\le K<3$, has image of doubling at most $2K-2$, and this factor is best possible. Further consequences include variants of the $3k-4$ theorem for popular sumsets and an inverse theorem for Tao’s convolution inequality.