  We use analytic conformal bootstrap methods to determine the anomalous
dimensions and OPE coefficients for large spin operators in general conformal
field theories in four dimensions containing a scalar operator of conformal
dimension $\Delta_\phi$. It is known that such theories will contain an
infinite sequence of large spin operators with twists approaching
$2\Delta_\phi+2n$ for each integer $n$. By considering the case where such
operators are separated by a twist gap from other operators at large spin, we
analytically determine the $n$, $\Delta_\phi$ dependence of the anomalous
dimensions. We find that for all $n$, the anomalous dimensions are negative for
$\Delta_\phi$ satisfying the unitarity bound. We further compute the first
subleading correction at large spin and show that it becomes universal for
large twist. In the limit when $n$ is large, we find exact agreement with the
AdS/CFT prediction corresponding to the Eikonal limit of a 2-2 scattering with
dominant graviton exchange.
