Restricted Families of Projections and Random Subspaces

Abstract

We study the restricted families of orthogonal projections in R 3 . We show that there are families of random subspaces which admit a Marstrand-Mattila type projection theorem.

Improved Bounds for Restricted Families of Projections to Planes in ℝ3

“…for z ∈ Z and j ∈ {1, 2, 3}. Consequently, 1) . (3.13) The rest of the argument is devoted to finding an upper bound for the left hand side of (3.13); comparing the bounds will complete the proof.…”

“…Theorem 1.1 improves on both results for 1 < dim H K ≤ 3 2 (being sharp in that range), and improves on (1.2) whenever dim H K < 9 4 . As a related development, we mention the recent paper of Chen [1], where the author constructs, for any α ∈ (1,2], an α-Ahlfors-David set G ⊂ S 2 such that (MM1)-(MM2) are valid for H α | G in place of H 2 . It seems likely that Chen's method also works with α = 1, if Ahlfors-David regularity is relaxed to 0 < H 1 (G) < ∞, but the resulting set G needs to be much more "uniformly distributed" than the circles S W (or, in fact, any other curves Γ ⊂ S 2 of finite length), see [1, Lemmas 2.1-2.2].…”

Projections in vector spaces over finite fields

“…For restricted families of projections in Euclidean spaces, the author [4] obtained that some random subsets of sphere of R 3 admit a Marstrand-Mattila type projection theorem. For more details, see [4].…”

Dimension Distortion by Right Coset Projections in the Heisenberg Group

“…Analogous, but in some cases weaker, results have been obtained when projections are restricted to a subfamily of planes [3,16,12,23,9,24,15]. In [25] the authors introduced the concept of transversal families of maps thus giving a vast generalization of Theorem 1.1 which extended the result to many more families of mappings.…”