Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 129 / 3 / ii

Apply the Croot-Sisask almost-periodicity theorem with the sampling set $A$, $S=G$, $f=1_{-A}$, and error $ϵ/k$. Since $A+G=G$, the doubling parameter is $K={\alpha }^{-1}$, and $\Vert 1_{-A}{\Vert }_p\le 1$. We obtain $X\subseteq G$ withsuch thatfor every $t\in X-X$. Every $x\in kX-kX$ is a sum of $k$ elements of $X-X$. Telescoping these $k$ shifts and using translation invariance and the triangle inequality for the Lp norm gives