We prove that the finite field Fourier extension operator for the paraboloidis bounded from $L^2o L^r$ for $rgeq rac{2d+4}{d}$ in even dimensions$dge 8$, which is the optimal $L^2$ estimate. For $d=6$ we obtain the optimalrange $r> rac{2d+4}{d}=8/3$, apart from the endpoint. For $d=4$ we improvethe prior range of $r>16/5=3.2$ to $rgeq 28/9=3.111ldots$, compared to theconjectured range of $rgeq3$. The key new ingredient is improved additiveenergy estimates for subsets of the paraboloid.
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机译：我们证明了Paraboloidis的有限字段傅里叶扩展运算符从$ l ^ 2 to l ^ r $ for $ r geq frac {2d + 4} {d} $ di尺寸$ d ge 8 $，这是最佳的$ l ^ 2 $估计。对于$ d = 6 $我们获得最佳r> frac {2d + 4} {d} = 8/3 $，除了端点。对于$ d = 4 $我们即将到来的$ r> 16/5 = 3.2 $至$ r geq 28/9 = 3.111 ldots $，与$ r geq3 $的标注范围相比。关键的新成分是改善抛物面的亚群的添加剂估计。
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