Extending the Reach of the Point-to-Set Principle

Lutz, Jack H.; Lutz, Neil; Mayordomo, Elvira

doi:10.2139/ssrn.4254623

Cited by 2 publications

(3 citation statements)

References 29 publications

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“…Lutz and Lutz [25] and Lutz and Stull [28] provide outlines and applications of the point-to-set principle. Recently, the point-to-set principle has been extended to arbitrary separable metric spaces [26].…”

Section: 4mentioning

confidence: 99%

Co-analytic Counterexamples to Marstrand's Projection Theorem

Richter¹

2023

Preprint

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Assuming V =L, we construct a plane set E of Hausdorff dimension 1 whose every orthogonal projection onto straight lines through the origin has Hausdorff dimension 0. This is a counterexample to J. M. Marstrand's seminal projection theorem [30]. While counterexamples had already been constructed decades ago, initially by R. O. Davies [8], the novelty of our result lies in the fact that E is co-analytic. Following Marstrand's original proof [30] (and R. Kaufman's newer, and now standard, approach [18] based on capacities), a counterexample to the projection theorem cannot be analytic, hence our counterexample is optimal. We then extend the result in a strong way: we show that for each ∈ (0, 1) there exists a co-analytic set E of dimension 1 + , each of whose orthogonal projections onto straight lines through the origin has Hausdorff dimension . The constructions of E and E are by induction on the countable ordinals, applying a theorem by Z. Vidnyánszky [42].

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Section: 4mentioning

confidence: 99%

Co-analytic Counterexamples to Marstrand's Projection Theorem

Richter¹

2023

Preprint

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“…Lutz and Lutz [91] and Lutz and Stull [95,96] provide applications of the point-to-set principle, which has also been extended to arbitrary separable metric spaces [92].…”

Section: Effective Notions In Fractal Geometrymentioning

confidence: 99%

“…This work has culminated in a recent identification of Hausdorff dimension in terms of algorithmic com-plexity of its individual points, the so-called point-to-set principle, due to Jack Lutz and Neil Lutz [91], which has proven to be immensely applicable [95,96,94,147]. It has also been extended to spaces beyond 2 ω [92].…”

Section: Chapter 1 Introductionmentioning

confidence: 99%

On the Definability and Complexity of Sets and Structures

Richter

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<p><strong>We prove results at the intersection of computability theory and set theory, broadly concerning notions of complexity in the sense of definability. We consider these in the contexts of problems in classical mathematics: in the first part of this thesis, we present the solution to a problem in uncountable computable structure theory concerning the construction of complicated uncountable free abelian groups, which was obtained in collaboration with Greenberg, Shelah, and Turetsky. In the second part, we turn towards fractal geometry and its connection with both descriptive set theory and algorithmic randomness. We construct a co-analytic set of reals which fails Marstrand’s Projection Theorem, a seminal result of classical fractal geometry and geometric measure theory. The construction uses computability-theoretical tools, in particular the notion of Kolmogorov complexity.</strong></p>

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Extending the Reach of the Point-to-Set Principle

Cited by 2 publications

References 29 publications

Co-analytic Counterexamples to Marstrand's Projection Theorem

Co-analytic Counterexamples to Marstrand's Projection Theorem

On the Definability and Complexity of Sets and Structures

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