  The purpose of this paper is to introduce and study some basic concepts of
quantitative rectifiability in the first Heisenberg group $\mathbb{H}$. In
particular, we aim to demonstrate that new phenomena arise compared to the
Euclidean theory, founded by G. David and S. Semmes in the 90's. The theory in
$\mathbb{H}$ has an apparent connection to certain nonlinear PDEs, which do not
play a role with similar questions in $\mathbb{R}^{3}$.
  Our main object of study are the intrinsic Lipschitz graphs in $\mathbb{H}$,
introduced by B. Franchi, R. Serapioni and F. Serra Cassano in 2006. We claim
that these $3$-dimensional sets in $\mathbb{H}$, if any, deserve to be called
quantitatively $3$-rectifiable. Our main result is that the intrinsic Lipschitz
graphs satisfy a weak geometric lemma with respect to vertical $\beta$-numbers.
Conversely, extending a result of David and Semmes from $\mathbb{R}^{n}$, we
prove that a $3$-Ahlfors-David regular subset in $\mathbb{H}$, which satisfies
the weak geometric lemma and has big vertical projections, necessarily has big
pieces of intrinsic Lipschitz graphs.
