Measurable sets with excluded distances

For a set of distances D = { d 1 , … , d k } a set A in the plane 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. We also show that there exists a largest D-avoiding set, and give an algorithm to compute the maximum density of a D-avoiding set for any D.