As mentioned in the bounty, I'm actually looking for a set $E$ such that $d(D)=[0,1]$. The original question follows for context.
Let $E \subset \mathbb{R}$ be Lebesgue measurable, let $D \subseteq \mathbb{R}$ the set of all points for which the Lebesgue density of $E$ exists, and let $d : D \to [0,1]$ denote the density.
Loosely speaking, my question is about how "large" $d(D) \subseteq [0,1]$ can be. Specifically, I'm wondering about the following two questions:
- Can $d(D)$ be uncountable?
- Can $d(D)$ have nonempty interior?
As long as I haven't made any simple mistakes thinking about the problem, then in higher dimensions, $d$ can be surjective, but the examples I've imagined only work because a line segment in $\mathbb{R}^n$ has measure zero for $n > 1$ (so they have no apparent parallel in $\mathbb{R}$ due to Lebesgue's density theorem).
For $\mathbb{R}$, at this point I've only done the "trivial" case, that $d(D)$ can contain any prespecified countable set as a subset, so in particular it can be dense in $[0,1]$. I don't even know if I have the tools to answer the questions I'm left with.