Dimension Spectra of Lines

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.

Algorithmic Fractal Dimensions in Geometric Measure Theory

“…N. Lutz and Stull [62] have also shown that the dimension spectrum of a line is always infinite, proving the following two results. The first theorem proves that if dim(a, b) = Dim(a, b) then the corresponding line contains a length one interval.…”

“…(N. Lutz and Stull[62]) Let a, b ∈ R satisfy that dim(a, b) = Dim(a, b). Then for every s ∈ [0, 1] there is a point x ∈ R such that dim(x, ax + b) = s + min{dim(a, b), 1}.The second result proves that all spectra of lines are infinite.Theorem 3.7.…”

ON A METRIC GENERALIZATION OF THE tt-DEGREES AND EFFECTIVE DIMENSION THEORY

“…In computability theory (particularly in algorithmic randomness theory), it is usual to consider the algorithmic dimension (the algorithmic information density) of a point in the context of fractal dimension; see [8,Section 13]. The notions of Kolmogorov complexity and algorithmic dimension in a Euclidean space has been studied in [24,23,26,25], for instance. In this section, we compare our notion of topological dimension of points and the notions of effective fractal dimension of points.…”

“…Given x ∈ R n , the Kolmogorov complexity of x at precision r (cf. [24,23,26,25]) is defined as follows:…”

Algorithmic randomness and analysis

“…In this chapter, we study the spectra of possible dimensions of points on a line in the Euclidean plane. This chapter is joint work with Neil Lutz and some portion of it have appeared in [33] and [34].…”