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 $d1+t$ 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 Cantortype sets whose graphs have large dimensions. As an application, we construct a collection $E$ of disjoint, nonparallel $k$planes in $\mathbb{R}^d$, for $d \geq k+2$, 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 arXiv:
 arXiv:1906.06288
 Bibcode:
 2019arXiv190606288C
 Keywords:

 Mathematics  Metric Geometry;
 28A78
 EPrint:
 The presentation of the construction and the argument in Section 4 was completely rewritten