Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 129 / 2 / v / Solution

Iterate part (iv) as a density increment. At a stage with ambient vector space of dimension $n_i$ and relative density ${\alpha }_i$, either the sumset contains a coset of a $k$-dimensional subspace, or there is a subspace of codimension $O({\alpha }_i^{-2}k)$ and a coset on which the density is at least $3{\alpha }_i/2$. Translate that coset back to the subspace. Since the ambient group has characteristic two, this translation does not alter the translated set's sumset.

The densities grow geometrically, so the iteration has $O(\log ({\alpha }^{-1}))$ stages, while the total codimension lost isChoose $k=\lfloor c{\alpha }^{2}n\rfloor$ with a sufficiently small absolute $c>0$. The total codimension is then less than $n-k$, so every stage still has ambient dimension at least $k$. The density cannot increase indefinitely beyond one; therefore the first alternative must occur. When $c{\alpha }^{2}n<1$, take the zero-dimensional subspace; this is the usual integer rounding implicit in the asymptotic dimension bound. Thusfor an absolute $C>0$.