On restricted families of projections in ℝ³

Abstract

ABSTRACT. We study projections onto non-degenerate one-dimensional families of lines and planes in R 3 . Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most 1/2-dimensional sets B ⊂ R 3 is typically preserved under one-dimensional families of projections onto lines. We improve the result by an ε, proving that if dim H B = s > 1/2, then the packing dimension of the projections is almost surely at least σ(s) > 1/2. For projections onto planes, we obtain a similar bound, with the threshold 1/2 replaced by 1. In the special case of self-similar sets K ⊂ R 3 without rotations, we obtain a full Marstrand type projection theorem for oneparameter families of projections onto lines. The dim H K ≤ 1 case of the result follows from recent work of M. Hochman, but the dim H K > 1 part is new: with this assumption, we prove that the projections have positive length almost surely. CONTENTS

Hausdorff dimension estimates for restricted families of projections in R3

“…Classical techniques, dating back as far as Kaufman's work [8] in 1968, can be used to show that the lower bounds in [7] and [6] (obtained without any curvature assumptions) are no longer sharp for the projections ρ θ . In [4], we verified the following proposition: The lower bound for dim π θ (B) holds without the curvature condition (1.2) and was already established in [7]. In contrast, the bounds in [7] and [6] give no information about dim ρ θ (B) in this situation (at least in case dim B ≤ 1) -the reason being example (I) above.…”

“…The main results in [4] were the verification of the first part of this conjecture for selfsimilar sets in R 3 without rotations, and a slight improvement over the min{dim B, 1/2} and min{dim B, 1} bounds for packing dimension dim p :…”

“…To formulate the hypothesis of the conjecture in precise terms, K. Fässler and the author proposed in [4] the following curvature condition for one-dimensional families of one-and two-planes in R 3 . Write θ := span(γ(θ)) ∈ G (3,1), and V θ := ⊥ θ ∈ G (3,2).…”

Fractal Geometry and Stochastics V

“…The conjectured lower bound is min{dim H E, 2} and the bound min{dim H E, 1} for all values of dim H E was obtained in [26]. The further improvements stated below come from Fourier restriction methods [65].…”

“…[26,65] Let E ⊂ R 3 be a Borel or analytic set, let θ(t) be a non-degenerate family of directions, and let proj V θ (t) denote projection onto the plane perpendicular to direction θ. Then, for almost all t ∈ (0, 1),…”