This question resisted attempts on MSE.

Let $E \subset \mathbb R^n$ be a Lebesgue measurable set. We say that $x \in \mathbb R^n$ is a turbulent point of E if both the following conditions hold:

$$\liminf_{r \to 0_+} \frac{|E \cap B_r (x)|}{|B_r (x)|} = 0$$

$$\limsup_{r \to 0_+}\frac{|E \cap B_r (x)|}{|B_r (x)|} = 1.$$

In other words, the density of $E$ oscillates maximally between $0$ and $1$.

How large can this set be? More specifically,

Question: What is the supremal Hausdorff dimension of the set of turbulent points of a subset of $\mathbb R^n$? And is the supremum achieved?

This question resisted attempts on MSE.

Let $E \subset \mathbb R^n$ be a Lebesgue measurable set. We say that $x \in \mathbb R^n$ is a turbulent point of E if both the following conditions hold:

$$\liminf_{r \to 0_+} \frac{|E \cap B_r (x)|}{|B_r (x)|} = 0$$

$$\limsup_{r \to 0_+}\frac{|E \cap B_r (x)|}{|B_r (x)|} = 1.$$

In other words, the density of $E$ oscillates maximally between $0$ and $1$.

How large can this set be? More specifically,

Question: What is the supremal Hausdorff dimension of the set of turbulent points of a subset of $\mathbb R^n$? And is the supremum achieved?