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

The sum over all $x$ of the threefold-convolution representation function is $|A{|}^3$. Some $x_0$ therefore satisfiesUse part c with $\epsilon =\alpha /2$. There is a vector subspace $V$ of codimension $O({\alpha }^{-2}\log (1/\alpha ))$ such that, for every $t\in V$,Positivity means that $x_0+t$ has a representation as a sum of three elements of $A$. Thuswhich is the desired translate of a low-codimension subspace.