  The Quadratic Density Hales Jewett conjecture with $2$ letters states that
for large enough $n$, every dense subset of $\{0,1\}^{n^{2}}$ contains a
combinatorial line where the wildcard set is of the form $\gamma \times \gamma$
where $\gamma \subset \{1,2,\dots n\}$. We show in an elementary quantitative
way that every dense subset of $\{0,1\}^{n^{2}}$, for sufficiently large $n$,
contains two elements such that the set of coordinate points where they differ,
which we term the difference set of these two elements, is of the form
$\gamma_{1}\times \gamma_{2}$ where $\gamma_1, \gamma_2$ are both nonempty
subsets of $\{1,2,\dots n\}$. Further we give several non-trivial examples of
dense vector subspaces of $\{0,1\}^{n^{2}}$, where in each case the wildcard
set of the combinatorial line that can be obtained has restrictions on its size
and shape.
