Measurable sets with excluded distances

2007/03/28 by Boris Bukh, Bukh, Boris
Mathematics · #05D10 #52C10 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #math.CA #math.CO #msc:05D10 #msc:52C10

paper · pdf · doi:10.48550/arxiv.math/0703856

23 pages, 3 figures, typos and small errors fixed

arxiv created 2008/02/24 · arxiv updated 2009/12/01

Abstract

For a set of distances D=d₁,...,d_k a set A is called D-avoiding if no pair of points of A is at distance d_i for some i. We show that the density of A is exponentially small in k provided the ratios d₁/d₂, d₂/d₃, ..., d_k-1/d_k are all small enough. This resolves a question of Szekely, and generalizes a theorem of Furstenberg-Katznelson-Weiss, Falconer-Marstrand, and Bourgain. Several more results on D-avoiding sets are presented.

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