= Injective linear modelling of a small sumset
{title2=$\ker L\cap12B=\{0\},\quad |G'|<4|12B|$}
For a finite subset $B\subseteq\mathbb F_2^N$ containing zero, choose $m=\lceil\log_2(2|12B|)\rceil$ and a uniformly random <linear map> $L:\mathbb F_2^N\to\mathbb F_2^m$. Each fixed nonzero vector has zero image with <probability> $2^{-m}$. The <union bound> gives a map whose <kernel> meets $12B$ only at zero. It is injective on $6B$, and $2^m<4|12B|$. The <Plünnecke-Ruzsa inequality> makes the image of $B$ have density at least $1/(4C^{12})$ if $|B+B|\le C|B|$.
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