= Isotropic-line obstruction to finite-field restriction
{title2=$R^*(2\to q)\geq p^{2/q-1/2}$}
If $i^2=-1$ in the prime field, the line $\ell=\{(t,it,0):t\in\mathbb F_p\}$ lies in the three-dimensional <discrete paraboloid>. Its indicator has normalized surface $L^2$ norm $p^{-1/2}$. Its extension is $p^{-1}$ on the two-dimensional annihilator plane and zero elsewhere. Thus its ambient $\ell^q$ norm is $p^{2/q-1}$, proving the displayed lower bound, which diverges for $q<4$.
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