Large sets with small injective projections

Frank Coen¹ ; Nate Gillman² ; Tamás Keleti³ ; Dylan King⁴ ; Jennifer Zhu⁵

¹ Villanova University, Department of Mathematics & Statistics
² Brown University, Department of Mathematics
³ Eötvös Loránd University, Institute of Mathematics
⁴ Wake Forest University, Department of Mathematics & Statistics
⁵ University of California, Department of Mathematics, Berkeley

Annales Fennici Mathematici, Tome 46 (2021) no. 2, pp. 683-702

Résumé

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.

Détail

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