The sum over all $x$ of the threefold-convolution representation function is $|A|^3$. Some $x_0$ therefore satisfies
$$
(1_A*1_A*1_A)(x_0)\geq\frac{|A|^3}{p^n}=\alpha|A|^2.
$$
Use part c with $\varepsilon=\alpha/2$. There is a <vector subspace> $V$ of codimension $O(\alpha^{-2}\log(1/\alpha))$ such that, for every $t\in V$,
$$
(1_A*1_A*1_A)(x_0+t)
\geq\frac\alpha2|A|^2>0.
$$
Positivity means that $x_0+t$ has a representation as a sum of three elements of $A$. Thus
$$
x_0+V\subseteq A+A+A,
$$
which is the desired translate of a low-codimension subspace.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 129 / 3 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-129/3/d/solution