  The Gross-Neveu model defines a unitary CFT of interacting fermions in
$2<d<4$ which has perturbative descriptions in the $1/N$ expansion and in the
epsilon-expansion near two and four dimensions. In each of these descriptions,
the CFT has an infinite tower of nearly conserved currents of all spins. We
determine the structure of the non-conservation equations both at large $N$ and
in the epsilon-expansion, and use it to find the leading order anomalous
dimensions of the broken currents. Similarly, we use the fact that the CFT
spectrum includes a nearly free fermion to fix the leading anomalous dimensions
of a few scalar composite operators. We also compute the scaling dimensions of
double-trace spinning operators in the large $N$ expansion, which correspond to
interaction energies of two-particle states in the AdS dual higher-spin theory.
We first derive these anomalous dimensions by a direct Feynman diagram
calculation, and then show that the result can be exactly reproduced by
analytic bootstrap methods, provided the sum over the tower of weakly broken
higher-spin currents is suitably regularized. Finally, we apply the analytic
bootstrap approach to derive the anomalous dimensions of the double-trace
spinning operators in the 3d bosonic and fermion vector models coupled to
Chern-Simons theory, to leading order in $1/N$ but exactly in the `t Hooft
coupling.
