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Geometric and Functional Analysis

, Volume 18, Issue 3, pp 668–697 | Cite as

Measurable Sets With Excluded Distances

  • Boris Bukh
Article

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.

Keywords and phrases:

Excluded distances Hadwiger-Nelson problem chromatic number of the plane measurable coloring distance graph chromatic number Euclidean Ramsey theory 

AMS Mathematics Subject Classification:

52C10 05D10 11B75 

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Copyright information

© Birkhaeuser 2008

Authors and Affiliations

  1. 1.Department of Mathematics, Fine HallPrincetonUSA

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