Limit laws in the lattice problem. II. The case of ovals

We study the error of the number of unimodular lattice points that fall into a dilated and centred ellipse around $0$. We first show that the study of the error, when the error is normalized by $\sqrt{t}$ with $t$ the parameter of dilatation of the ellipse, when $t$ tends to infinity and when the lattice is random, is reduced to the study of a Siegel transform $\mathcal{S}(f_{t})(L)$ that depends on $t$. Then, by making $t \rightarrow \infty$, we see that $\mathcal{S}(f_{t})$ converges in law towards a modified Siegel transform with random weights $\mathcal{S}(F)(\theta,L)$ where $\theta$ is a second random parameter. Finally, we show that this last quantity converges almost surely and we study the existence of the moments of its law.