Combinatorial proofs of two theorems of Lutz and Stull

Abstract

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⊂Rn is any set with equal Hausdorff and packing dimensions, then dimHπe(K)=min{dimHK,1} for almost everye ∈Sn−1. Here π estands for orthogonal projection to 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 generalise Lutz and Stull’s theorems: the versions in this paper apply to orthogonal projections tom-planes in Rn, for all 0