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].
@InProceedings{stull:LIPIcs.ICALP.2022.133, author = {Stull, D. M.}, title = {{The Dimension Spectrum Conjecture for Planar Lines}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {133:1--133:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.133}, URN = {urn:nbn:de:0030-drops-164749}, doi = {10.4230/LIPIcs.ICALP.2022.133}, annote = {Keywords: Algorithmic randomness, Kolmogorov complexity, effective dimension} }
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