AbstractI prove that closed n-regular sets $$E \subset {\mathbb {R}}^{d}$$
E
⊂
R
d
with plenty of big projections have big pieces of Lipschitz graphs. In particular, these sets are uniformly n-rectifiable. This answers a question of David and Semmes from 1993.
We show that, aside from the semiring equations, three equations and two equation schemes characterize the semiring of regular sets with the Kleene star operation.