Dimension Spectra of Lines

2017/01/15 by Neil Lutz, Lutz, Neil, D. M. Stull +1 · 2 citations
Computer Science · #Cellular Automata and Applications #Computability, Logic, AI Algorithms #Computational Complexity (cs.CC) #Digital Image Processing Techniques #FOS: Computer and information sciences #cs.CC

paper · pdf · doi:10.48550/arxiv.1701.04108

arxiv created 2017/01/15 · openalex publication_date 2017/01/15 · arxiv updated 2017/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp(L) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim(a, b) is equal to the effective packing dimension Dim(a, b), then sp(L) contains a unit interval. We also show that, if the dimension dim(a, b) is at least one, then sp(L) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.

Cited by

• Algorithmic Information Bounds for Distances and Orthogonal Projections
• Lines in Every Direction with No ee-Random Points

Related

• The Dimension Spectrum Conjecture for Planar Lines
• Bounding the Dimension of Points on a Line
• Projection Theorems Using Effective Dimension
• Dimension Functions on the Spectrum over Bounded Geodesics and Applications to Diophantine Approximation
• Some spectral applications of McMullen's Hausdorff dimension algorithm