Measurable sets with excluded distances

For a set of distances D={d1,…,dk} a set A in the plane 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. 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.