What is the Lebesgue density of $A$ and $B$ which answers a previous question?

Suppose we partition the reals into two sets $A$ and $B$ that are dense (with positive Lebesgue measure) in every non-empty sub-interval $(a,b)$ of $\mathbb{R}$, where Lebesgue measure $\lambda$ restricts outer measure $\lambda^{*}$ to sets measurable in the Caratheodory sense.

For $A$ and $B$ which answers this question, if $\mathbf{B}_{\varepsilon}(x)$ is a closed ball of radius $\varepsilon$ centered at $x\in\mathbb{R}$, what is: $$d_{\varepsilon}(x)=\frac{\lambda(A\cap \mathbf{B}_{\varepsilon}(x))}{2\varepsilon}$$ and $$d_{\varepsilon}^{\,*}(x)=\frac{\lambda(B\cap \mathbf{B}_{\varepsilon}(x))}{2\varepsilon}$$ in terms of $\varepsilon$ and $x$?

  • Paul F Paul F

    I accepted your question so we can communicate here. Do you want to post this question also in MSE?

  • Paul F Paul F

    Done: https://math.stackexchange.com/questions/4750841/what-is-the-lebesgue-density-sets-a-and-b

    • You forgot to include a link to the first question (include the first one on math stack and the link to this question).

    • Paul F Paul F

      What is the link. Please provide it here.

    • Here's the link to the first question: https://math.stackexchange.com/q/4750001/125918

    • You can also include this post.

  • Paul F Paul F

    Done. But it it the link to the question which is currently closed.


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  • According to the same user on Reddit (https://www.reddit.com/r/mathematics/comments/15lzpev/what_is_the_asymptotic_density_and_lebesgue/jvlsx4w/?utm_source=share&utm_medium=ios_app&utm_name=ioscss&utm_content=1&utm_term=1&context=3), they realized the answer is wrong and this answer (https://math.stackexchange.com/a/4747177/125918) needs to be adjusted.

  • Paul I found the true answer to the first question here: https://math.stackexchange.com/a/4754256/125918 ; however, I need to answer it with this question.

The answer is accepted.
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