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The solid-angle sum $A_{\mathcal{P}} (t)$ of a rational polytope${\mathcal{P}} \subset \mathbb{R}^d$, with $t \in \mathbb{Z}$ was firstinvestigated by I.G. Macdonald. Using our Fourier-analytic methods, we are ableto establish an explicit formula for $A_{\mathcal{P}} (t)$, for any realdilation $t$ and any rational polygon ${\mathcal{P}} \subset \mathbb{R}^2$. Ourformulation sheds additional light on previous results, for lattice-pointenumerating functions of triangles, which are usually confined to the case ofinteger dilations. Our approach differs from that of Hardy and Littlewood in1992, but offers an alternate point of view for enumerating weighted latticepoints in real dilations of real triangles.

Macdonald's solid-angle sum for real dilations of rational polygons