Large Sets with Small Injective Projections

Abstract

Let ${\ell }_1,{\ell }_2,\dots$ be a countable collection of lines in ${\mathbb{R}}^d$. For any $t\in [0,1]$ we construct a compact set $\Gamma \subset {\mathbb{R}}^d$ with Hausdorff dimension $d-1+t$ which projects injectively into each ${\ell }_i$, such that the image of each projection has dimension $t$. This immediately implies the existence of homeomorphisms between certain Cantor-type sets whose graphs have large dimensions. As an application, we construct a collection $E$ of disjoint, non-parallel $k$-planes in ${\mathbb{R}}^d$, for $d\ge k+2$, whose union is a small subset of ${\mathbb{R}}^d$, either in Hausdorff dimension or Lebesgue measure, while $E$ itself has large dimension. As a second application, for any countable collection of vertical lines $w_i$ in the plane we construct a collection of nonvertical lines $H$, so that $F$, the union of lines in $H$, has positive Lebesgue measure, but each point of each line $w_i$ intersects at most one $h\in H$ and, for each $w_i$, the Hausdorff dimension of $F\cap w_i$ is zero.

The presentation of the construction and the argument in Section 4 was completely rewritten

References in corpus (2)

• Small unions of affine subspaces and skeletons via Baire category
• A "rare'' plane set with Hausdorff dimension 2