Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 129 / 3 / b / Solution

Apply the assumed finite-field character approximation with error $ϵ/2$. We obtain $k\le 4q/{ϵ}^2$ characters ${\gamma }_i$ and a functionsuch that $\Vert f-P{\Vert }_{L^q}\le (ϵ/2)\Vert \hat{f}{\Vert }_{{\ell }^1}$. LetEach character has a kernel of codimension at most one, so $\mathrm{codim}W\le k\le 4q/{ϵ}^2$. For $x\in W$, ${\tau }_{x}P=P$, and translation invariance of the $L^q$ norm gives

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