Bounding the Dimension of Points on a Line

Lutz, Neil; Stull, Donald M.

doi:10.1007/978-3-319-55911-7_31

Cited by 35 publications

(66 citation statements)

References 26 publications

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“…This follows from the point-to-set principle of J. Lutz and N. Lutz [9] and the existence of Furstenberg sets with parameter α and Hausdorff dimension less than 1 + α (attributed by Wolff [16] to Furstenberg and Katznelson "in all probability"). The argument is simple and very similar to our proof in [11] of a lower bound on the dimension of generalized Furstenberg sets. Specifically, for every s ∈ [0, 1], we want to find an x of effective Hausdorff dimension s such that (1) holds.…”

Section: Background and Approachsupporting

confidence: 60%

“…It was shown by Turetsky that, for every n ≥ 2, the set of all points in R n with effective Hausdorff 1 is connected, guaranteeing that 1 ∈ sp(L a,b ). In recent work [11], we showed that the dimension spectrum of a line in R 2 cannot be a singleton. By proving a general lower bound on dim(x, ax + b), which is presented as Theorem 5 here, we demonstrated that min{1, dim(a, b)} + 1 ∈ sp(L a,b ) .…”

Section: Introductionmentioning

confidence: 97%

“…These dimensions are both algorithmically and geometrically meaningful [4]. In particular, the quantities sup x∈E dim(x) and sup x∈E Dim(x) are closely related to classical Hausdorff and packing dimensions of a set E ⊆ R n [6,9], and this relationship has been used to prove nontrivial results in classical fractal geometry using algorithmic information theory [13,9,11].…”

Section: Introductionmentioning

confidence: 99%

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Dimension Spectra of Lines

Lutz

Stull

2017

Unveiling Dynamics and Complexity

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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.

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Section: Background and Approachsupporting

confidence: 60%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Dimension Spectra of Lines

Lutz

Stull

2017

Unveiling Dynamics and Complexity

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“…In [10] it was proved that if F ⊂ R 2 is an (α, s)-Furstenberg set, then dim F ≥ 2α − 1 + s and dim F ≥ α + s 2 . In [8] Lutz and Stull investigated the generalized Furstenberg-problem using methods from information theory. They proved that if F ⊂ R 2 is an (α, s)-Furstenberg set, then dim F ≥ α + min{s, α}.…”

Section: Introduction and Statements Of The Main Resultsmentioning

confidence: 99%

Hausdorff dimension of Furstenberg-type sets associated to families of affine subspaces

Héra¹

2019

Ann. Acad. Sci. Fenn. Math.

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We show that if B ⊂ R n and E ⊂ A(n, k) is a nonempty collection of k-dimensional affine subspaces of R n such that every P ∈ E intersects B in a set of Hausdorff dimension at least α with k − 1 < α ≤ k, then dim B ≥ α + dim E/(k + 1), where dim denotes the Hausdorff dimension. This estimate generalizes the well known Furstenberg-type estimate that every α-Furstenberg set in the plane has Hausdorff dimension at least α + 1/2.More generally, we prove that if B and E are as above with 0 < α ≤ k, then dim B ≥ α + (dim E − (k − α )(n − k))/( α + 1). We also show that this bound is sharp for some parameters.As a consequence, we prove that for any 1 ≤ k < n, the union of any nonempty s-Hausdorff dimensional family of k-dimensional affine subspaces of R n has Hausdorff dimension at least k + s k+1 .

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

Section: 22mentioning

confidence: 99%