  We study the free part of the Bernoulli action of $\mathbb{Z}^n$ for $n\geq
2$ and the Borel combinatorics of the associated Schreier graphs. We construct
orthogonal decompositions of the spaces into marker sets with various
additional properties. In general, for Borel graphs $\Gamma$ admitting weakly
orthogonal decompositions, we show that $\chi_B(\Gamma)\leq 2\chi(\Gamma)-1$
under some mild assumptions. As a consequence, we deduce that the Borel
chromatic number for $F(2^{\mathbb{Z}^n})$ is $3$ for all $n\geq 2$. Weakly
orthogonal decompositions also give rise to Borel unlayered toast structures.
We also construct orthogonal decompositions of $F(2^{\mathbb{Z}^2})$ with
strong topological regularity, in particular with all atoms homeomorphic to a
disk. This allows us to show that there is a Borel perfect matching for
$F(2^{\mathbb{Z}^n})$ for all $n\geq 2$ and that there is a Borel lining of
$F(2^{\mathbb{Z}^2})$.
