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 that, in (MM1)-(MM2), the 2-dimensional measure H 2 on S 2 can be replaced by length measure on certain curves Γ ⊂ S 2. The main result in [5] proved this for part (MM1), whenever Γ is a circle, but not a great circle (the great circles are a "degenerate" case, having non-trivial orthogonal complement). We refer the reader to [5] for a broader introduction, and earlier results, on the projections ρ e. In this paper, we consider part (MM2) in the same setting: Theorem 1.1. Let W ⊂ R 3 be a 2-plane, which is not a subspace. If K ⊂ R 3 is a Borel set, then dim H π e (K) ≥ min dim H K, 1 + dim H K 3 for H 1 almost every e on the circle S W = S 2 ∩ W. In particular, the projections π e , e ∈ S W , preserve H 1 almost surely the dimension of at most 3 2-dimensional Borel sets.

Restricted families of projections onto planes: the general case of nonvanishing geodesic curvature

“…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.…”

Hausdorff dimension and projections related to intersections

“…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].…”

An a.e. lower bound for Hausdorff dimension under vertical projections in the Heisenberg group

“…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…”

“…(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.…”