Finite field restriction estimates for the paraboloid in high even dimensions

We prove that the finite field Fourier extension operator for the paraboloid is bounded from $L^2\to L^r$ for $r\ge \frac{2d+4}{d}$ in even dimensions $d\ge 8$, which is the optimal $L^2$ estimate. For $d=6$ we obtain the optimal range $r>\frac{2d+4}{d}=8\mathrm{/}3$, apart from the endpoint. For $d=4$ we improve the prior range of $r>16\mathrm{/}5=3.2$ to $r\ge 28\mathrm{/}9=3.111\dots$, compared to the conjectured range of $r\ge 3$. The key new ingredient is improved additive energy estimates for subsets of the paraboloid.