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Bounds on the dimension of lineal extensions

Ryan E. G. Bushling, Jacob B. Fiedler

TL;DR

This paper investigates how the dimension of a union of line segments changes when each segment is extended to its containing line (lineal extension). In the plane, it establishes a strong packing-dimension result and derives a corollary for the generalized Kakeya problem, then provides partial higher-dimensional bounds via Besicovitch-set estimates. The core method blends effective techniques from Kolmogorov complexity with the point-to-set principle to obtain dimension bounds, including a sharp Hausdorff-bound for s-Hausdorff extensions and a packing-bound equality for 1-Hausdorff extensions. The results connect line-segment extension problems to Furstenberg-type sets and Kakeya-type questions, yielding new quantitative bounds and illustrating intriguing pathological behaviors of s-packing extensions. Collectively, the work advances understanding of how geometric extension interacts with fractal dimensions and offers tools potentially applicable to Kakeya-type phenomena in higher dimensions.

Abstract

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, from which the generalized Kakeya conjecture for packing dimension immediately follows. This is followed by several doubling estimates in higher dimensions and connections to related problems.

Bounds on the dimension of lineal extensions

TL;DR

This paper investigates how the dimension of a union of line segments changes when each segment is extended to its containing line (lineal extension). In the plane, it establishes a strong packing-dimension result and derives a corollary for the generalized Kakeya problem, then provides partial higher-dimensional bounds via Besicovitch-set estimates. The core method blends effective techniques from Kolmogorov complexity with the point-to-set principle to obtain dimension bounds, including a sharp Hausdorff-bound for s-Hausdorff extensions and a packing-bound equality for 1-Hausdorff extensions. The results connect line-segment extension problems to Furstenberg-type sets and Kakeya-type questions, yielding new quantitative bounds and illustrating intriguing pathological behaviors of s-packing extensions. Collectively, the work advances understanding of how geometric extension interacts with fractal dimensions and offers tools potentially applicable to Kakeya-type phenomena in higher dimensions.

Abstract

Let be a union of line segments and the set obtained from by extending each line segment in to a full line. Keleti's line segment extension conjecture posits that the Hausdorff dimension of should equal that of . Working in , we use effective methods to prove a strong packing dimension variant of this conjecture, from which the generalized Kakeya conjecture for packing dimension immediately follows. This is followed by several doubling estimates in higher dimensions and connections to related problems.
Paper Structure (14 sections, 24 theorems, 105 equations, 1 figure)

This paper contains 14 sections, 24 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Line segment extension in R2
  3. Line segment extension in Rn and elementary Besicovitch set estimates
  4. Background on the line segment extension conjecture and its relatives
  5. History and context
  6. s-Hausdorff extensions and the Furstenberg set conjecture
  7. Preliminaries on effective methods
  8. Basic definitions
  9. Some useful results
  10. Line segment extension in the plane
  11. The Hausdorff dimension bound for s-Hausdorff extensions
  12. The packing dimension bound for 1-Hausdorff extensions
  13. Pathological behavior of s-packing extensions
  14. Extension in higher dimensions and related problems

Key Result

Theorem 1.1

If $E \subseteq \mathbb{R}^2$ is a union of line segments, then $\mathop{\mathrm{\dim_{\mathrm{H}}}}\nolimits \mathrm{\mathbf{L}}(E) = \mathop{\mathrm{\dim_{\mathrm{H}}}}\nolimits E$.

Figures (1)

  • Figure 1: (i) The definition of $c_r$ guarantees that the teal property holds on $[1,c_r]$. (ii) We choose $D$ such that the teal property holds on $[1,r]$ relative to $(A,D)$and the average growth rate of $K_s^{A,D}(a,b)$ on $[c_r,r]$ is close to $1$.

Theorems & Definitions (30)

  • Conjecture 1: Line segment extension conjecture
  • Theorem 1.1: Keleti keleti2016lines
  • Definition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 1.5
  • Theorem 1.6: Keleti keleti2016lines
  • Lemma 1.7
  • Corollary 1.8
  • ...and 20 more