This question is based on some Physics motivation. Define a distance function $f(\mathbf{r})=\int_{\Omega }d^2k\int_{\Omega }d^2q \cos[(\mathbf{k}-\mathbf{q})\cdot\mathbf{r}]$, where $\mathbf{r},\mathbf{k},\mathbf{q}\in \mathbb{R}^2$ and $\Omega \subset\mathbb{R}^2$ is some finite region.

The question is: At large distance $\left | \mathbf{r} \right |\rightarrow \infty $, what's the asymptotic behavior of this function $f(\mathbf{r})$? Does it decay like $e^{-\left | \mathbf{r} \right |/L}$ or $\left | \mathbf{r} \right |^{-a}$(with some characteristic length $L$ and some positive number $a$)? Or does the large distance behavior depend on the geometry of the region $\Omega$?

Thank you very much.