More precisely:
Question: Let $\gamma \colon [0,1] \to \mathbb{R}^2$ be Lipschitz. Do there exist Borel (or Suslin) sets $A_i \subset \mathbb{R}^2$ and directions $v_i \in \mathbb{R}^2$, for every $i \in \mathbb{N}$, such that
- $\gamma([0,1]) = \bigcup_{i \in \mathbb{N}}A_i$ and
- the projections $p_i \colon A_i \to \mathbb{R} \colon x \mapsto x \cdot v_i$ are injective for all $i \in \mathbb{N}$?
What I know is that (obviously) every Lipschitz curve can be covered by countably many Lipschitz graphs up to $\mathcal{H}^1$-measure zero, but that there also exist Lipschitz curves that cannot be covered by countably many Lipschitz graphs. So, the question is how much we have to relax the requirement of having Lipschitz graphs in order to cover everything.
If the answer to my question is affirmative, it might simplify a proof I am currently working on. The answer is not crucial for the proof, but the question itself seems interesting enough (for me) to be asked here.