Ultra-large distance modification of gravity from Lorentz symmetry breaking at the Planck scale

We present an extension of the Randall--Sundrum model in which, due to spontaneous Lorentz symmetry breaking, graviton mixes with bulk vector fields and becomes quasilocalized. The masses of KK modes comprising the four-dimensional graviton are naturally exponentially small. This allows to push the Lorentz breaking scale to as high as a few tenth of the Planck mass. The model does not contain ghosts or tachyons and does not exhibit the van Dam--Veltman--Zakharov discontinuity. The gravitational attraction between static point masses becomes gradually weaker with increasing of separation and gets replaced by repulsion (antigravity) at exponentially large distances.