Orthogonal projections of planar sets in countable directions

Given $0<t\le 1$ and $C>0$. For any $t/2\le s<t$, we prove that there exists a finite subset $\Omega\subset S^1$, such that for all $t$-Ahlfors-David regular set $E\subset [0,1)^2$ with regular constant at most $C$, we have $$ \max_{e\in \Omega} \overline{\dim}_B \pi_e(E)\ge s, $$ where $\pi_e(E)$ stands for the orthogonal projection of $E$ into the line spanned by $e\in S^1$. This was known earlier, due to Orponen (2016), for the situation $t=1$. As an application of our results, I show that there exists a countable subset $\Omega\subset S^1$ such that for any $t$-Ahlfors-David regular set $E\subseteq [0,1)^2$, with $0<t\le 2$, one has $$ \sup_{e\in \Omega} \overline{\dim}_B \pi_e(E)=\min\{t, 1\}. $$ Moreover, for $0<t\le 1$ there exists $t$-Ahlfors-David regular set such that the supremum is not attained.
Comment: 18 pages