2018
DOI: 10.1093/imrn/rny193
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Improved Bounds for Restricted Families of Projections to Planes in ℝ3

Abstract: R 3 be any 2-plane, which is not a subspace. We prove that if K ⊂ R 3 is a Borel set with dimH K ≤ 3 2 , then dimH πe(K) = dimH K for H 1 almost every e ∈ S 2 ∩W , where H 1 denotes the 1-dimensional Hausdorff measure and dimH the Hausdorff dimension. This was known earlier, due to Järvenpää, Järvenpää, Ledrappier and Leikas, for Borel sets K with dimH K ≤ 1. We also prove a partial result for sets with dimension exceeding 3/2, improving earlier bounds by D. Oberlin and R. Oberlin. Recent evidence suggests tha… Show more

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Cited by 17 publications
(26 citation statements)
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References 11 publications
(23 reference statements)
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“…This partially resolves Conjecture 1.6 from [3] in the range dim A ≤ 3/2, for projections onto planes. In this range, this was previously known in the special case of non-great circles; due to Orponen and Venieri [14]. In the range 3/2 < dim A < 5/2, Theorem 1.1 improves and generalises the previously best known lower bound from [5], which was also specific to non-great circles.…”
Section: Introductionmentioning
confidence: 53%
“…This partially resolves Conjecture 1.6 from [3] in the range dim A ≤ 3/2, for projections onto planes. In this range, this was previously known in the special case of non-great circles; due to Orponen and Venieri [14]. In the range 3/2 < dim A < 5/2, Theorem 1.1 improves and generalises the previously best known lower bound from [5], which was also specific to non-great circles.…”
Section: Introductionmentioning
confidence: 53%
“…The family S g , g ∈ O(n), is a restricted family of orthogonal projections onto n-planes in R 2n ; it is only n(n − 1)/2-dimensional, while the full family of orthogonal projections has dimension n 2 . Similar questions for other restricted families of orthogonal projections have been studied by many people; see [17], [16], [11], [36], [34], [35], [21], [37]. There are also discussions on these in [30] and [32].…”
mentioning
confidence: 76%
“…Most of this section is devoted to proving the lemmas from which Theorem 1.1 will follow. The first lemma of this section is an abstract version of Lemma 2.5 from [18] (see also [14,Theorem 7.2]); the proof is not too different from the Euclidean case, but is included for completeness. In the statement of the lemma, (θ, x) → π θ (x) is an arbitrary continuous function, but all statements following the proof of the lemma will specialise to the case where…”
Section: Proof Of Lemmas and The Main Theoremmentioning
confidence: 99%
“…(the upper bound is roughly 3.2). The proof employs some of the techniques used by Orponen and Venieri in [18] for restricted families of projections in R 3 ; the main difficulty in adapting this to the Heisenberg setting lies in finding a substitute for Marstrand's "Three Circles Lemma" (see [20,Lemma 3.2]). The key observation is that if two points (z, t), (ζ, τ ) ∈ H have their vertical projections close to each other for some angle θ, then the second component |t − τ − 2z ∧ ζ| of the Korányi distance between them is small, and the latter function is independent of the angle θ.…”
Section: Introductionmentioning
confidence: 99%