Dimension Spectra Of Lines

Document Type

Article

Publication Date

5-1-2022

Published In

Computability

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 is at least one, then sp (L) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.

Keywords

Effective dimension, Kolmogorov complexity

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