So I've seen this question asked in the context of $\mathbb{R}$, but I'm unsure as to how I'm supposed to approach this problem in $\mathbb{R}^2$.
Furthermore, I was informed that there exists pretty easy examples of such sets in $\mathbb{R}^2$, but I'm unaware of any. The main issue I'm having in $\mathbb{R}^2$ is computing the measure of the intersection $E \cap B_r(x)$ (as this gets rather complicated quickly outside of pretty simple sets $E$). Ideally I want a set $E$ such that $m(E \cap B_r(x)) = a \pi r^2$ for small $r > 0$.
Does anyone have some suggestions on where to start? Thank you.
I seem to have overcomplicated this. Since I want $m(E \cap B_r(x)) = \alpha \pi r^2$, it's clear to me now that if you sweep out a section of the plane with angle $\theta = 2 \pi \alpha$, then you get exactly: $$\frac{m(E \cap B_r(x))}{m(B_r(x))} = \frac{\frac{2 \pi \alpha}{2 \pi}\pi r^2}{\pi r^2} = \alpha$$ using the formula for the area of the section of a circle (ofcourse this is at $x = (0,0)$).