Bounds on the dimension of lineal extensions Article Swipe

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Ryan E. G. Bushling , Jacob B. Fiedler ·

YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.4171/jfg/161

Let E \subseteq \mathbb{R}^{n} be a union of line segments and F \subseteq \mathbb{R}^{n} the set obtained from E by extending each line segment in E to a full line. Keleti’s line segment extension conjecture posits that the Hausdorff dimension of F should equal that of E . Working in \mathbb{R}^{2} , we use effective methods to prove a strong packing dimension variant of this conjecture. Furthermore, a key inequality in this proof readily entails the planar case of the generalized Kakeya conjecture for packing dimension. This is followed by several doubling estimates in higher dimensions and connections to related problems.

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Mathematics

Combinatorics

Planar Transmission Line

Geometry

Mathematical Analysis

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Conjecture Mathematics Hausdorff dimension Dimension (graph theory) Packing dimension Combinatorics Extension (predicate logic) Line (geometry) Planar Line segment Hausdorff space Minkowski–Bouligand dimension Geometry Mathematical analysis Computer science Fractal dimension Fractal Computer graphics (images) Programming language

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Type: article
Language: en
Landing Page: https://doi.org/10.4171/jfg/161
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DOI: https://doi.org/10.4171/jfg/161

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Title: Bounds on the dimension of lineal extensions

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Publication year: 2025

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Authors: Ryan E. G. Bushling, Jacob B. Fiedler

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Concepts: Conjecture, Mathematics, Hausdorff dimension, Dimension (graph theory), Packing dimension, Combinatorics, Extension (predicate logic), Line (geometry), Planar, Line segment, Hausdorff space, Minkowski–Bouligand dimension, Geometry, Mathematical analysis, Computer science, Fractal dimension, Fractal, Computer graphics (images), Programming language

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