Distance-Avoiding Sets in the Plane

Fix a real number d > 0. Let D = {1, d} if d 6= 1; otherwise D = {1} may simply be written as 1. A subset S ⊆ R is said to avoid D if kx− yk / ∈ D for all x, y ∈ S. For example, the union of open balls of radius 1/2 with centers in (2Z) avoids the distance 1. If instead the balls have centers in (3Z), then their union avoids {1, 2}. It is natural to ask about the “largest possible” S that avoids D. Let BR denote the ball of radius R with center 0. Assuming S is Lebesgue measurable, its density