Heisenberg Hausdorff Dimensionof Besicovitch Sets

Venieri, Laura

doi:10.2478/agms-2014-0013

Cited by 3 publications

(5 citation statements)

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“…Remark 14.5. In [21] in the Heisenberg group we found the same lower bound n+4 2 for the Hausdorff dimension of bounded Kakeya sets when n ≤ 8. Moreover, we derived a better lower bound 4n+10 7 for n ≥ 9 from the Kakeya estimate obtained by Katz and Tao ( [14]) using arithmetic methods (see also Remark 12.1).…”

Section: Moreover By (122)supporting

confidence: 61%

“…Note that in the proof of Lemma 14.8 we showed that Axiom 2 holds also with balls defined with respect to d ∞ and tubes defined with respect to the Euclidean metric in the case of a Carnot group of step 2 and m 2 = 1 (see ( 123)). Thus we could prove the following, which for Heisenberg groups is the same as Theorem 1 in [21] (and it can be proved as Theorem 4.1). Let G be a Carnot group of step 2 with m 2 = 1, let 1 ≤ p < n, β > 0 such that n + 1 − βp > 0.…”

Section: Axiommentioning

confidence: 80%

“…Axiom 2: In [21] we considered (bounded) Kakeya sets in the Heisenberg group, which we can identify with R n = R 2N ×R, equipped with the Korányi metric d H (which is bi-Lipschitz equivalent to the Carnot-Carathéodory metric). The proof of Theorem 1 in [21] contains the proof that Axiom 2 holds (with θ = 0) when the tubes are defined with respect to the Euclidean metric and the balls with respect to d H . We could use essentially the same proof also for tubes defined with respect to d H .…”

Section: Axiommentioning

confidence: 99%

“…One motivation for this last example comes from the idea of studying Kakeya sets in Carnot groups. The author in [21] has proved that L p estimates for the classical Kakeya maximal function imply lower bounds for the Hausdorff dimension of bounded Besicovitch sets in the Heisenberg group H n ∼ = R 2n+1 with respect to the Korányi metric (which is bi-Lipschitz equivalent to the Carnot Carathéodory metric). By the results of Wolff and of Katz and Tao one then gets the lower bounds 2n+5 2 for n ≤ 3 and 8n+14 7 for n ≥ 4 for the Heisenberg Hausdorff dimension.…”

Section: Examples Of Applications 27 1 Introductionmentioning

confidence: 99%

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Dimension estimates for Kakeya sets defined in an axiomatic setting

Venieri¹

2017

Ann. Acad. Sci. Fenn. Math. Diss.

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In this dissertation we define a generalization of Kakeya sets in certain metric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesgue measure containing a segment of length one in every direction. A famous conjecture, known as Kakeya conjecture, states that the Hausdorff dimension of any Kakeya set should equal the dimension of the space. It was proved only in the plane, whereas in higher dimensions both geometric and arithmetic combinatorial methods were used to obtain partial results.In the first part of the thesis we define generalized Kakeya sets in metric spaces satisfying certain axioms. These allow us to prove some lower bounds for the Hausdorff dimension of generalized Kakeya sets using two methods introduced in the Euclidean context by Bourgain and Wolff. With this abstract setup we can deal with many special cases in a unified way, recovering some known results and proving new ones.In the second part we present various applications. We recover some of the known estimates for the classical Kakeya and Nikodym sets and for curved Kakeya sets. Moreover, we prove lower bounds for the dimension of sets containing a segment in a line through every point of a hyperplane and of an (n-1)-rectifiable set. We then show dimension estimates for Furstenberg type sets (already known in the plane) and for the classical Kakeya sets with respect to a metric that is homogeneous under non-isotropic dilations and in which balls are rectangular boxes with sides parallel to the coordinate axis. Finally, we prove lower bounds for the classical bounded Kakeya sets and a natural modification of them in Carnot groups of step two whose second layer has dimension one, such as the Heisenberg group. On the other hand, if the dimension is bigger than one we show that we cannot use this approach.

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Section: Moreover By (122)supporting

confidence: 61%

Section: Axiommentioning

confidence: 80%

Section: Axiommentioning

confidence: 99%

Section: Examples Of Applications 27 1 Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dimension estimates for Kakeya sets defined in an axiomatic setting

Venieri¹

2017

Ann. Acad. Sci. Fenn. Math. Diss.

Self Cite

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“…The Heisenberg Hausdorff dimension of Euclidean Kakeya sets has been studied in [16] where the author showed a lower bound on the dimension of Kakeya sets. In [17], the author studied Kakeya sets for general metric spaces in axiomatic sense.…”

Section: Definition 11mentioning

confidence: 99%