Measurable sets with excluded distances
Abstract
For a set of distances D=d1,...,dk a set A is called D-avoiding if no pair of points of A is at distance di for some i. We show that the density of A is exponentially small in k provided the ratios d1/d2, d2/d3, ..., dk-1/dk 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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