  We derive forms of light-state dominance for correlators in CFT$_d$, making
precise the sense in which correlators can be approximated by the contribution
of light operator exchanges. Our main result is that the four-point function of
operators with dimension $\Delta$ is approximated, with bounded error, by the
contribution of operators with scaling dimension below $\Delta_c > 2\Delta$ in
the appropriate OPE channel. Adapting an existing modular invariance argument,
we use crossing symmetry to show that the heavy-state contribution is
suppressed by a relative factor of $e^{2\Delta-\Delta_c}$. We extend this
result to the first sheet and derivatives of the correlator. Further exploiting
technical similarities between crossing and modular invariance, we prove
analogous results for the $2d$ partition function along the way.
  We then turn to effective field theory in gapped theories and AdS/CFT, and
make some general comments about the effect of integrating out heavy particles
in the bulk. Combining our bounds with the Lorentzian OPE inversion formula we
show that, under certain conditions, light-state dominance implies that
integrating out heavy exchanges leads to higher-derivative couplings suppressed
at large $\Delta_{gap}$.
