  Atoms deeply trapped in magic wavelength optical lattices provide a Doppler-
and collision-free dense ensemble of quantum emitters ideal for high precision
spectroscopy. Thus, they are the basis of some of the best optical clock setups
to date. However, despite their minute optical dipole moments the inherent long
range dipole-dipole interactions in such lattices generate line shifts,
dephasing and modified decay. We show that in a perfectly filled lattice these
effects are resonantly enhanced depending on lattice constant, lattice geometry
and excitation scheme inducing clock shifts of many atomic linewidths and
reducing measurement precision via superradiance. However, under optimal
conditions collective effects can be exploited to yield zero effective shifts
and prolong dipole lifetimes beyond the single atom decay. In particular we
identify 2D hexagonal or square lattices with lattice constants below the
optical wavelength as most promising configurations for an accuracy and
precision well below the independent ensemble limit. This geometry should also
be an ideal basis for related applications such as superradiant lasers,
precision magnetometry or long lived quantum memories.
