Optimal local laws and CLT for the circular Riesz gas

We study the long-range one-dimensional Riesz gas on the circle, a continuous system of particles interacting through a Riesz (i.e inverse power) kernel. We establish near-optimal rigidity estimates on gaps valid at any scale. Leveraging on these local laws and using a Stein method, we provide a quantitative Central Limit Theorem for linear statistics. The proof is based on a mean-field transport and on a fine analysis of the fluctuations of local error terms through the study of Helffer-Sj\"ostrand equations. The method can handle very singular test-functions, including characteristic functions of intervals, using a comparison principle for the Helffer-Sj\"ostrand equation