  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 to be Borel, or even analytic. One of the
theorems states that if $K \subset \mathbb{R}^{n}$ is any set with equal
Hausdorff and 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 for orthogonal projection to $\mathrm{span}(e)$.
  The primary purpose of this paper is to present proofs for Lutz and Stull's
projection theorems which do not refer to information theoretic concepts.
Instead, they will rely on combinatorial-geometric arguments, such as
discretised versions of Kaufman's "potential theoretic" method, the pigeonhole
principle, and a lemma of Katz and Tao. A secondary purpose is to slightly
generalise Lutz and Stull's theorems: the versions in this paper apply to
orthogonal projections to $m$-planes in $\mathbb{R}^{n}$, for all $0 < m < n$.
