A simple proof of the Bourgade-Sodin-Zeitouni theorem on the excess of the sine-$\beta$ process for non-integer $\sqrt{2/\beta}>1$

arXiv:2611.14655v1 Announce Type: new Abstract: We give an alternative proof of the Bourgade-Sodin-Zeitouni theorem on the excess of the sine-$\beta$ process for the Paley-Wiener space when $\sqrt{2/\beta}\in (1,+\infty)$ is not an integer. Let $k(\beta)=\lfloor\sqrt{2/\beta}\rfloor$. For almost every configuration, removing any $k(\beta)$ points leaves a minimal uniqueness set for the Paley-Wiener space. The same conclusion holds for each Bernstein space $\mathcal{B}^p$ of $L^p(\mathbb{R})$-integrable functions with $m(\beta)<p<\infty$, where $m(\beta)=\beta(k(\beta)+1)-\sqrt{\beta^2(k(\beta)+1)^2-2\beta}$.

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