Hausdorff dimension estimates for restricted families of projections in R3

Abstract

This paper is concerned with restricted families of projections in R 3 . Let K ⊂ R 3 be a Borel set with Hausdorff dimension dim K = s > 1. If G is a smooth and sufficiently wellcurved one-dimensional family of two-dimensional subspaces, the main result states that there exists σ(s) > 1 such that dim π V (K) ≥ σ(s) for almost all V ∈ G. A similar result is obtained for some specific families of one-dimensional subspaces.

Application of a Fourier Restriction Theorem to Certain Families of Projections in $${\mathbb {R}}^3$$ R 3

“…One of the problems treated in [4] is to say something about the dimension of π t (B) for generic t ∈ [0, 1]. Fässler and Orponen prove that (a) if α ≤ 1 then dim(π t (B)) = dim (B) for almost all t ∈ [0, 1], and (b) if α > 1 then there exists σ = σ(α) > 1 such that the packing dimension of π t (B) exceeds σ for almost all t. In a subsequent paper, [6], Orponen considers the particular γ given by…”

Sixty Years of Fractal Projections

“…Orponen [71] also showed that there exist numbers σ(λ) > 1 defined for λ > 1, and increasing with λ, such that if dim H E > 1 then dim…”

“…Estimates for packing dimensions of projections may be found in [26]. The introduction of the paper [71] provides a recent overview of this area.…”

Geometry and Analysis of Fractals

“…In [Orp3] Orponen was able to do this for the special family of orthogonal projections onto the lines l θ spanned by (cos θ, sin θ, 1) and their orthogonal complements. In [Orp3] Orponen was able to do this for the special family of orthogonal projections onto the lines l θ spanned by (cos θ, sin θ, 1) and their orthogonal complements.…”