2022
DOI: 10.1090/proc/15914
|View full text |Cite
|
Sign up to set email alerts
|

On the dimension of Kakeya sets in the first Heisenberg group

Abstract: We define Kakeya sets in the Heisenberg group and show that the Heisenberg Hausdorff dimension of Kakeya sets in the first Heisenberg group is at least 3. This lower bound is sharp since, under our definition, the { x o y } \{xoy\} -plane is a Kakeya set with Heisenberg Hausdorff dimension 3.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
2

Citation Types

0
5
0

Year Published

2023
2023
2023
2023

Publication Types

Select...
2
1

Relationship

0
3

Authors

Journals

citations
Cited by 3 publications
(5 citation statements)
references
References 16 publications
(17 reference statements)
0
5
0
Order By: Relevance
“…Proof of Theorem Without loss of generality, we may assume that scriptL=false(Pfalse)$\mathcal {L} = \ell (P)$ is a Borel set of lines, that is, PR3$P \subset \mathbb {R}^{3}$ is a Borel set. For the full details of this reduction, see [7, Section 3] or [1, Theorem 7.9]. The idea is that we can first replace scriptL$\cup \mathcal {L}$ by a Gδ$G_{\delta }$‐set GscriptL$G \supset \cup \mathcal {L}$ without affecting dimH(L)$\dim _{\mathrm{H}}(\cup \mathcal {L})$.…”
Section: Proofs Concerning Horizontal Linesmentioning
confidence: 99%
See 3 more Smart Citations
“…Proof of Theorem Without loss of generality, we may assume that scriptL=false(Pfalse)$\mathcal {L} = \ell (P)$ is a Borel set of lines, that is, PR3$P \subset \mathbb {R}^{3}$ is a Borel set. For the full details of this reduction, see [7, Section 3] or [1, Theorem 7.9]. The idea is that we can first replace scriptL$\cup \mathcal {L}$ by a Gδ$G_{\delta }$‐set GscriptL$G \supset \cup \mathcal {L}$ without affecting dimH(L)$\dim _{\mathrm{H}}(\cup \mathcal {L})$.…”
Section: Proofs Concerning Horizontal Linesmentioning
confidence: 99%
“…This interval would (be chosen to) consist of points ydouble-struckR$y \in \mathbb {R}$ with the property that the plane double-struckWy$\mathbb {W}_{y}$ intersects a family of segments corresponding to a false(dimHPεfalse)$(\dim _{\mathrm{H}}P - \epsilon )$‐dimensional Borel subset PP$P^{\prime } \subset P$. We refer the reader to [7, Section 3] for a very similar argument. Clearly, (3.1) follows from (3.2) by the ‘Fubini inequality’ for Hausdorff measures (hence dimension), see [1, Theorem 5.8] or [8, Theorem 7.7].…”
Section: Proofs Concerning Horizontal Linesmentioning
confidence: 99%
See 2 more Smart Citations
“…They introduced a method of point-plate incidences, and proved this in the case dim A = 3 by using a square function estimate for the cone of Guth, Wang and Zhang [9] to control the average L 2 norms of pushforwards of discretised 3-dimensional measures. The point-line duality principle they used is due to Liu [10].…”
Section: Introductionmentioning
confidence: 99%