Measurable sets with excluded distances

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 Sz´ekely, and generalizes a theorem of Furstenberg-Katznelson-Weiss, Falconer-Marstrand, and Bourgain. Several more results on D-avoiding sets are presented.

• Type: Journal
• Year: 2007
• Where: ENDM
• Authors: Boris Bukh
• Keywords: Distance Di, ENDM 2007, Ratios, Small