Inverse Theorems for Point-Sphere Incidences over Finite Fields

Let Q be a nondegenerate quadratic form on $\mathbb F_q^d$, with q odd & $d\ge2$. We prove a two-scale inverse theory for point-sphere incidences. At the maximal-incidence endpoint, put $μ=1-I(P,S)/(|P||S|)$. If $μ<(d+1)/(d+2)^2$, then deleting at most a proportion $\sqrt{(d+1)μ}$ from each side p