We consider the unique continuation properties of asymptotically
Anti-de Sitter spacetimes by studying Klein-Gordon-type equations gφ +
σφ = G(φ, ∂φ), σ ∈ R, on a large class of such spacetimes. Our main result
establishes that if φ vanishes to sufficiently high order (depending on σ) on
a sufficiently long time interval along the conformal boundary I, then the
solution necessarily vanishes in a neighborhood of I. In particular, in the
σ-range where Dirichlet and Neumann conditions are possible on I for the
forward problem, we prove uniqueness if both these conditions are imposed.
The length of the time interval can be related to the refocusing time of null
geodesics on these backgrounds and is expected to be sharp. Some global
applications as well a uniqueness result for gravitational perturbations are
also discussed. The proof is based on novel Carleman estimates established in
this setting.