The Dimension Spectrum Conjecture for Planar Lines

2021/01/30 by D. M. Stull, Donald M. Stull, Stull, D. M. · 2 citations
Computer Science · Mathematics · #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #Computational Complexity (cs.CC) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Computer and information sciences #FOS: Mathematics #cs.CC #math.CO

paper · pdf · doi:10.48550/arxiv.2102.00134

openalex publication_date 2021/01/30 · arxiv created 2021/11/04 · arxiv updated 2021/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let L_a,b be a line in the Euclidean plane with slope a and intercept b. The dimension spectrum \spec(L_a,b) is the set of all effective dimensions of individual points on L_a,b. The dimension spectrum conjecture states that, for every line L_a,b, the spectrum of L_a,b contains a unit interval. In this paper we prove that the dimension spectrum conjecture is true. Let (a,b) be a slope-intercept pair, and let d = min\dim(a,b), 1\. For every s ∈ (0, 1), we construct a point x such that dim(x, ax + b) = d + s. Thus, we show that \spec(L_a,b) contains the interval (d, 1+ d). Results of Turetsky , and Lutz and Stull, show that \spec(L_a,b) contain the endpoints d and 1+d. Taken together, [d, 1 + d] ⊆ \spec(L_a,b), for every planar line L_a,b.

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