Affiliations: [a] Computer Science Department, Swarthmore College, Swarthmore, PA 19081, USA. lutz@cs.swarthmore.edu | [b] Department of Computer Science, Northwestern University, Evanston, IL 60208, USA. donald.stull@northwestern.edu
Note: [1] A preliminary version of a portion of this work was presented at Unveiling Dynamics and Complexity, the 13th Conference on Computability in Europe (CiE 2017), Turku, Finland, June 12–16, 2017.
Note: [*] Research supported in part by National Science Foundation Grants 1247051 and 1545028.
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.
