 Solution

Let $T=1_A\ast 1_A$ and chooseApply part b with its error parameter replaced by a sufficiently small absolute multiple of $\epsilon$. This produces a vector subspace $V$ of codimensionwith the evident harmless modification when $\alpha =1$.

For $t\in V$, $x\in {\mathbb{F}}_p^n$, and $q=2m$, Hölder's inequality givesSince ${\alpha }^{-1/q}\le e^{1/2}$ by the choice of $m$, the bound from part b is at mostafter absorbing the absolute factor into the chosen error parameter. This is the required uniform estimate for $1_A\ast 1_A\ast 1_A$.