Combinatorial proofs of two theorems of Lutz and Stull
Orponen, T. (2021). Combinatorial proofs of two theorems of Lutz and Stull. Mathematical proceedings of the Cambridge Philosophical Society, 171(3), 503514. https://doi.org/10.1017/S0305004120000328
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2021Copyright
© Cambridge Philosophical Society 2021
Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrandtype 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 combinatorialgeometric 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 tomplanes in Rn, for all 0
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https://converis.jyu.fi/converis/portal/detail/Publication/51784739
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The author was supported by the Academy of Finland via the projects Quantitative rectifiability in Euclidean and nonEuclidean spaces and Incidences on Fractals, grant Nos. 309365, 314172, 321896.License
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