Hausdorff Dimension, Projections, Intersections, and Besicovitch Sets

Mattila, Pertti

doi:10.1007/978-3-030-32353-0_6

Cited by 15 publications

(14 citation statements)

References 71 publications

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“…The projections {π t } t∈R k were first investigated by Oberlin, see [18]. His results imply the following theorem, see also [16].…”

Section: And the Claim Concludes Takingmentioning

confidence: 95%

A Fubini-type theorem for Hausdorff dimension

Héra¹,

Keleti²,

Máthé³

2021

Preprint

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It is well known that a classical Fubini theorem for Hausdorff dimension cannot hold; that is, the dimension of the intersections of a fixed set with a parallel family of planes do not determine the dimension of the set. Here we prove that a Fubini theorem for Hausdorff dimension does hold modulo sets that are small on all Lipschitz curves/surfaces.We say thatwhere ess-sup(dim Et) is the essential supremum of the Hausdorff dimension of the vertical sections {Et} t∈R k of E, assuming that proj R k E has positive measure.We also obtain more general results by replacing R k by an Ahlfors regular set. Applications of our results include Fubini-type results for unions of affine subspaces and related projection theorems.

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“…The projections {π t } t∈R k were first investigated by Oberlin, see [18]. His results imply the following theorem, see also [16].…”

Section: And the Claim Concludes Takingmentioning

confidence: 95%

A Fubini-type theorem for Hausdorff dimension

Héra¹,

Keleti²,

Máthé³

2021

Preprint

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“…Chapter 24] and the survey [16]. Our main interest is in restricted (d, k)-sets, which we introduce now, and in estimates for the Fourier dimension.…”

Section: Kakeya Sets (D K)-sets and Dimension Theorymentioning

confidence: 99%

On the Fourier dimension of (d, k)-sets and Kakeya sets with restricted directions

Fraser

Kroon

2022

Math. Z.

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A (d, k)-set is a subset of $$\mathbb {R}^d$$ R d containing a k-dimensional unit ball of all possible orientations. Using an approach of D. Oberlin we prove various Fourier dimension estimates for compact (d, k)-sets. Our main interest is in restricted (d, k)-sets, where the set only contains unit balls with a restricted set of possible orientations $$\Gamma $$ Γ . In this setting our estimates depend on the Hausdorff dimension of $$\Gamma $$ Γ and can sometimes be improved if additional geometric properties of $$\Gamma $$ Γ are assumed. We are led to consider cones and prove that the cone in $$\mathbb {R}^{d+1}$$ R d + 1 has Fourier dimension $$d-1$$ d - 1 , which may be of interest in its own right.

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“…The average behaviour of Hausdorff dimension under orthogonal projections in Euclidean space was first explored by Marstrand in 1954 [15]; many developments and generalisations have occurred since (see e.g. [16,19,10,3,17]). An effort began in [1] and [2] towards understanding the behaviour of Hausdorff dimension under projections in the Heisenberg group, which was further developed in [12] and [11] (see also [5]).…”

Section: Introductionmentioning

confidence: 99%