eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-28
133:1
133:20
10.4230/LIPIcs.ICALP.2022.133
article
The Dimension Spectrum Conjecture for Planar Lines
Stull, D. M.
1
Department of Computer Science, Northwestern University, Evanston, IL, USA
Let L_{a,b} be a line in the Euclidean plane with slope a and intercept b. The dimension spectrum sp(L_{a,b}) is the set of all effective dimensions of individual points on L_{a,b}. Jack Lutz, in the early 2000s posed the dimension spectrum conjecture. This 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. Specifically, 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 sp(L_{a,b}) contains the interval [d, 1+ d].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol229-icalp2022/LIPIcs.ICALP.2022.133/LIPIcs.ICALP.2022.133.pdf
Algorithmic randomness
Kolmogorov complexity
effective dimension