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Plane Geometry is about flat shapes like lines, circles and triangles , shapes that can be drawn on a piece of paper

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When do quasiperiodic functions on $\mathbb{R}^2$ have an average over the plane?

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 $ …