Large sets with small injective projections


  • Frank Coen Villanova University, Department of Mathematics & Statistics
  • Nate Gillman Brown University, Department of Mathematics
  • Tamás Keleti Eötvös Loránd University, Institute of Mathematics
  • Dylan King Wake Forest University, Department of Mathematics & Statistics
  • Jennifer Zhu University of California, Department of Mathematics, Berkeley


Hausdorff dimension, Lebesgue measure, injective projections, union of lines, union of disjoint planes


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.



How to Cite

Coen, F., Gillman, N., Keleti, T., King, D., & Zhu, J. (2021). Large sets with small injective projections. Annales Fennici Mathematici, 46(2), 683–702. Retrieved from