Optimal asymptotic expansion of entire solutions to Monge-Ampère equation with $C^{\alpha }$ perturbed periodic data

We consider the asymptotic behavior at infinity of solution $u$ to Monge-Ampère equation $\det (D^{2}u)=f$ in $\text{rn}$, where $f$ is a perturbation of a periodic function and is only assumed to be Hölder continuous, compared to the previous work that $f$ is at least $C^{1,\text{az}}$. The consequence established in this paper, by a nonlocal method, is that the difference between $u$ and a quadratic polynomial is asymptotically close to a periodic function.