Measurable Sets With Excluded Distances |
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Authors: | Boris Bukh |
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Institution: | (1) Department of Mathematics, Fine Hall, Washington Road, Princeton, NJ 08544, USA |
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Abstract: | For a set of distances D = {d
1,..., 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 d
1/d
2, d
2/d
3, …, d
k-1/d
k
are all small enough. This resolves a question of Székely, and generalizes a theorem of Furstenberg–Katznelson–Weiss, Falconer–Marstrand,
and Bourgain. Several more results on D-avoiding sets are presented.
Received: January 2007, Revision: February 2008, Accepted: February 2008 |
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Keywords: | and phrases:" target="_blank"> and phrases: Excluded distances Hadwiger-Nelson problem chromatic number of the plane measurable coloring distance graph chromatic number Euclidean Ramsey theory |
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