Combinatorial proofs of two theorems of Lutz and Stull

Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed be Borel, or even analytic. One the theorems states that if $K \subset \mathbb{R}^{n}$ is any set with equal Hausdorff packing dimensions, then $$ \dim_{\mathrm{H}} \pi_{e}(K) = \min\{\dim_{\mathrm{H}} K,1\} for almost every $e \in S^{n - 1}$. Here $\pi_{e}$ stands orthogonal $\mathrm{span}(e)$. The primary purpose this paper present proofs Stull's do refer theoretic concepts. Instead, they will rely on combinatorial-geometric arguments, such as discretised versions Kaufman's "potential theoretic" method, pigeonhole principle, a lemma Katz Tao. A secondary slightly generalise theorems: in apply projections $m$-planes $\mathbb{R}^{n}$, all $0 < m n$.