On sets containing a unit distance in every direction

2019/12/03 by Shmerkin, Pablo, Yu, Han
#28A80 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Primary 05D99

paper · doi:10.48550/arxiv.1912.01523

Abstract

We investigate the box dimensions of compact sets in ℝ² that contain a unit distance in every direction (such sets may have zero Hausdorff dimension). Among other results, we show that the lower box dimension must be at least (4)/(7) and can be as low as (2)/(3). This quantifies in a certain sense how far the unit circle is from being a difference set.

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