A dichotomy for projections of planar sets

TL;DR

This paper establishes a dichotomy for projections of planar discrete sets, showing that most directions yield either dense or discrete projections, with a small set of exceptions in measure but not necessarily in Hausdorff dimension.

Contribution

It proves that the set of directions where the dichotomy fails is measure zero but can have large Hausdorff dimension, extending understanding of projection behaviors.

Findings

Most projections are either dense or discrete

Exceptional directions form a measure-zero set

The exceptional set can have large Hausdorff dimension

Abstract

We prove that most one-dimensional projections of a discrete subset of a plane are either dense in R (the real line), or form a discrete subset of R. More precisely, the set E of exceptional directions (for which the indicated dichotomy fails) is a meager subset of the unit circle T of Lebesgue measure 0. The set E however does not need to be small in the sense of Hausdorff dimension.