# Large Sets with Small Injective Projections

## TL;DR

This paper constructs complex fractal sets with prescribed projection properties, demonstrating the existence of high-dimensional sets with small or controlled projections, and applying these to geometric and measure-theoretic problems.

## Contribution

It introduces a method to build compact sets with specified Hausdorff dimensions that project injectively into lines, and applies this to construct sets with unusual measure and dimension properties.

## Key findings

- Existence of compact sets with prescribed projection dimensions
- Construction of small unions of high-dimensional planes with large overall dimension
- Examples of line collections with positive measure but limited intersection properties

## 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 $d \geq k+2$, 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.