Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 129 / 1 / d / Solution

Let $e_1,\dots ,e_m$ be the standard basis of ${\mathbb{F}}_2^m$ and takeIts sumset consists of zero, the $m$ basis vectors, and the $\left(\genfrac{}{}{0ex}{}{m}{2}\right)$ sums of two distinct basis vectors. ThusIf a coset $x+H$ contains $A_m$, then every difference of two elements of $A_m$ lies in $H$. In particular every $e_i$ lies in $H$, so $H={\mathbb{F}}_2^m$ andSince $m=2K_m+O(1)$, this ratio is eventually much larger than $K_m^{-1}{2}^{K_m}$. Hence no bound valid for all $K$ can replace the factor in part c by a function smaller than $K^{-1}{2}^K$.