Projections of planar sets in well-separated directions

First, let $K\subset B(0,1)\subset {\mathbb{R}}^2$ be a set with ${\mathcal{H}}_{\mathrm{\infty }}^1(K)\sim 1$, and write ${\pi }_e(K)$ for the orthogonal projection of $K$ into the line spanned by $e\in S^1$. For $1\mathrm{/}2\le s<1$, write $$E_s:=\{e:N({\pi }_e(K),\delta )\le {\delta }^{-s}\},$$ where $N(A,r)$ is the $r$-covering number of the set $A$. It is well-known – and essentially due to R. Kaufman – that $N(E_{s},\delta) \lessapprox \delta^{-s}$\mathrm{.}Usingthepolynomialmethod,Iprovethat$$ N(E_{s},r) \lessapprox \min\left{\delta^{-s}\left(\frac{\delta}{r}\right)^{1/2},r^{-1}\right}, \quad \delta \leq r \leq 1.I construct examples showing that the exponents in the bound are sharp for $\delta \leq r \leq \delta^{s}$. The second theorem concerns projections of $1$-Ahlfors-David regular sets. Let $A \geq 1$ and $1/2 \leq s < 1$ be given. I prove that, for $p = p(A,s) \in \mathbb{N}$ large enough, the finite set of unit vectors $S_{p} := \{e^{2\pi i k/p} : 0 \leq k < p\}$ has the following property. If $K \subset B(0,1)$ is non-empty and $1$-Ahlfors-David regular with regularity constant at most $A$, then\frac{1}{p} \sum_{e \in S_{p}} N(\pi_{e}(K),\delta) \geq \delta^{-s}$$forallsmallenough$\delta > 0$\mathrm{.}Inparticular,$\overline{\dim}{$\text{B}$} \pi{e}(K) \geq s$forsome$e \in S_{p}$.