in the paper, ON KAKEYA MAPS WITH REGULARITY ASSUMPTIONS, the author of the paper consider the Kakeya conjecture with some regularity on Kakeya map,

Kekeya conjecture[Hausdorff dimension version] If a set $K \subset \mathbb{R}^{n}$ contains a unit line segment in each direction (such a set is called a Kakeya set), then $K$ has Hausdorff dimension $n$.

Definition of Kakeya map. For any function $c: S^{n-1} \rightarrow \mathbb{R}^{n},$ we consider the associated map $\phi_{c}$ $$ \phi_{c}: S^{n-1} \times[0,1] \rightarrow \mathbb{R}^{n}, \quad(v, t) \mapsto c(v)+t v $$ We call $\phi_{c}$ a Kakeya map, and $K_{c}:=\operatorname{Im}\left(\phi_{c}\right)=\bigcup_{v \in S^{n-1}} c(v)+[0,1] \cdot v$ the associated Kakeya set.

Then the authors of this paper claim the following theorem:

Theorem1. If $c: \overline{B^{n-1}(0,1)} \rightarrow \mathbb{R}^{n-1}$ is $\alpha$ - Hölder continuous for some $\alpha>\frac{(n-2) n}{(n-1)^{2}},$ then $\operatorname{Im}\left(\phi_{c}\right)$ has positive Lebesgue measure.

Theorem2. If $c: \overline{B^{n-1}(0,1)} \rightarrow \mathbb{R}^{n-1}$ is continuous and in $H^{s}\left(\overline{B^{n-1}(0,1)}\right)$ for some $s>(n-1) / 2$, then $\operatorname{Im}\left(\phi_{c}\right)$ has positive Lebesgue measure.

The proof of the paper use winder number and isoperimetric inequality. But there is no reference for the previous work in this direction, that is, add some regularity on Kakeya conjecture and try to go further.

The problem is,

Problem 1: are there some similar papers in this direction?

and one more problem is,

Problem 2: why when adding $L^1_{loc}$ condition on Kakeya map, then for any given open set of $\mathbb{R}^n$ the preimage in $S^{n-1}$ is a measurable set, then it gives some Stickiness on different scales on the discretization problem, in this case, Kakeya problem is still hard? It seems we can get some more information on the multi-scale induction automatically.