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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

  1. $\gamma([0,1]) = \bigcup_{i \in \mathbb{N}}A_i$ and
  2. 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.

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    $\begingroup$ Are you still interested in this question? No clue how to solve it, but I could think about it. Have you asked Pertti or Marianna? I am a new user of this website trying to make an impact. The area of analysis is not active enough. Please ask more questions. $\endgroup$ Mar 26, 2018 at 21:50
  • $\begingroup$ @PiotrHajlasz Hi Piotr, this question came up when I was thinking about doubling measures and graphs of functions with Tuomo Ojala, arxiv.org/abs/1406.4693 (see also the improvement by Rossi and Shmerkin arxiv.org/abs/1702.01130 ). I have not thought about these things for a while and don't have a other motivation than pure curiosity currently for this questions. $\endgroup$ Mar 28, 2018 at 9:32

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