Notation: $\mu$ denotes the Lebesgue measure. Let $\mathcal D$ be the set of Lebesgue measurable subsets of $\mathbb R$ such that itself and its complement have nonzero Lebesgue measure in every interval.

We define the thinness $T_D (x)$ of a set $E \subset \mathbb R$ at $x \in \mathbb R$ by

$$T_D (x) := \sup \{r > 0 \,| \, \limsup_{\delta \to 0_+} \frac{\mu(D \cap B_d (x))}{\delta^r} < \infty.\},$$

and the overall thinness $T_D$ by

$$T_D = \sup_{x \in \mathbb R} T_D (x). $$

Question: What is the infimum, over $D \in \mathcal D$ of the greater of the thinness of $D, D^c$? That is, what is the value of the quantity

$$\inf_{D \in \mathcal D} \max(T_D, T_{D^c})?$$

Notation: $\mu$ denotes the Lebesgue measure. Let $\mathcal D$ be the set of Lebesgue measurable subsets of $\mathbb R$ such that itself and its complement have nonzero Lebesgue measure in every interval.

We define the thinness $T_D (x)$ of a set $D \subset \mathbb R$ at $x \in \mathbb R$ by

$$T_D (x) := \sup \{r > 0 \,| \, \limsup_{\delta \to 0_+} \frac{\mu(D \cap B_d (x))}{\delta^r} < \infty.\},$$

and the overall thinness $T_D$ by

$$T_D = \sup_{x \in \mathbb R} T_D (x). $$

Question: What is the infimum, over $D \in \mathcal D$ of the greater of the thinness of $D, D^c$? That is, what is the value of the quantity

$$\inf_{D \in \mathcal D} \max(T_D, T_{D^c})?$$