Large sets with small injective projections

Abstrakti

Let \(\ell_1,\ell_2,\dots\) be a countable collection of lines in \(\mathbf{R}^d\). For any \(t \in [0,1]\) we construct a compact set \(\Gamma\subseteq\mathbf{R}^d\) 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 \(\mathbf{R}^d\), for \(d \geq k+2\), whose union is a small subset of \(\mathbf{R}^d\), 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 \(w_i\) is contained in at most one \(h\in H\) and, for each \(w_i\), the Hausdorff dimension of \(F\cap w_i\) is zero.