  We prove an improvement of flatness result for nonlocal phase transitions.
For a class of nonlocal equations that includes $(-\Delta)^{s/2} u = u-u^3$,
with~$s\in(0,1)$, we obtain a result in the same spirit of a celebrated theorem
of Savin for the equation $-\Delta u = u-u^3$. As a consequence, we deduce that
entire solutions to~$(-\Delta)^{s/2} u = u-u^3$ with asymptotically flat level
sets are $1$D when~$s\in(0,1)$.
  The results presented are completely new even for the case of the fractional
Laplacian, but the robustness of the proofs allows us to treat also more
general, possibly anisotropic, integro-differential operators.
