A1 Journal article (refereed)
Combinatorial proofs of two theorems of Lutz and Stull (2021)

Orponen, T. (2021). Combinatorial proofs of two theorems of Lutz and Stull. Mathematical proceedings of the Cambridge Philosophical Society, 171(3), 503-514. https://doi.org/10.1017/S0305004120000328

JYU authors or editors

Publication details

All authors or editors: Orponen, Tuomas

Journal or series: Mathematical proceedings of the Cambridge Philosophical Society

ISSN: 0305-0041

eISSN: 1469-8064

Publication year: 2021

Publication date: 15/02/2021

Volume: 171

Issue number: 3

Pages range: 503-514

Publisher: Cambridge University Press (CUP)

Publication country: United Kingdom

Publication language: English

DOI: https://doi.org/10.1017/S0305004120000328

Publication open access: Not open

Publication channel open access: 

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/74553

Publication is parallel published: https://arxiv.org/abs/2002.01743

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

Keywords: measure theory; fractals; combinatorics

Free keywords: Hausdorff and packing measures

Contributing organizations

Ministry reporting: Yes

Reporting Year: 2021

JUFO rating: 2