Improvement of flatness for nonlocal phase transitions

We prove an improvement of flatness result for nonlocal phase transitions. For a class of nonlocal equations that includes $(-\mathrm{\Delta }{)}^{s\mathrm{/}2}u=u-u^3$, with~$s\in (0,1)$, we obtain a result in the same spirit of a celebrated theorem of Savin for the equation $-\mathrm{\Delta }u=u-u^3$. As a consequence, we deduce that entire solutions to~$(-\mathrm{\Delta }{)}^{s\mathrm{/}2}u=u-u^3$ 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.