A dichotomy for projections of planar sets

2012/03/03 by Boshernitzan, Michael
#Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT) #latex

paper · doi:10.48550/arxiv.1203.0669

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.

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