For example, consider the following
\begin{equation} \lim_{T\rightarrow\infty}\frac{1}{(2T)^2}\int^T_{-T}\int^T_{-T}\big[\cos(v/a)-\cos(u/b)\big]\cos(\sqrt{u^2+v^2})\ du\ dv, \end{equation}
where $a^2+b^2=1$ and $a,b>0$. The periodicity in the radial direction only coincides with that in the Cartesian directions along four rays (which incidentally make the integrand vanish).
Is the limit therefore nonexistent? Would any choice of the shape of integration region or discretization of $T$ allow the limit to exist as a function of $a$ (and/or $b$)? Are there any references that address similar problems in $\mathbb{R}^n$?
