The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group $\Bbb{H}$. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean theory, founded by G. David and S. Semmes in the 1990s. The theory in $\Bbb{H}$ has an apparent connection to certain nonlinear PDEs, which do not play a role with similar questions in ${\Bbb R}^3$.

Our main object of study are the {\it intrinsic Lipschitz graphs} in $\Bbb{H}$, introduced by B. Franchi, R. Serapioni, and F. Serra Cassano in 2006. We claim that these $3$-dimensional sets in $\Bbb{H}$, if any, deserve to be called quantitatively $3$-rectifiable. Our main result is that the intrinsic Lipschitz graphs satisfy a {\it weak geometric lemma} with respect to {\it vertical $\beta$-numbers}. Conversely, extending a result of David and Semmes from ${\Bbb R}^n$, we prove that a $3$-Ahlfors-David regular subset in $\Bbb{H}$, which satisfies the weak geometric lemma and has {\it big vertical projections}, necessarily has {\it big pieces of intrinsic Lipschitz graphs}.


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pp. 1087-1147
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