Dimension spectra of lines1

Author:

Lutz Neil1,Stull D.M.2

Affiliation:

1. Computer Science Department, Swarthmore College, Swarthmore, PA 19081, USA. lutz@cs.swarthmore.edu

2. Department of Computer Science, Northwestern University, Evanston, IL 60208, USA. donald.stull@northwestern.edu

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.

Publisher

IOS Press

Subject

Artificial Intelligence,Computational Theory and Mathematics,Computer Science Applications,Theoretical Computer Science

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