Polynomial-exponent constructions beat the grid's n^(1+o(1))
The argument
Because no scaling of the n-point square grid has more than n^(1+O(1/log log n)) unit distances, while there are n-point sets with more than n^(1.014) unit distances for arbitrarily large n and, by an independent route, the Lee-Pohoata-Zhu sets have a distance repeated about n^(1+δ) times, the ratio of the best known count to the grid's count tends to infinity along an infinite sequence of n, so the grid is not asymptotically optimal. Equivalently, the claim follows from the refutation of the conjectured bound u(n) ≤ n^(1+o(1)), which was the proposition that the grid's growth rate is essentially best possible.
Does it work?
Not yet evaluated.
Premises
- UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →The n-point square grid, however scaled, has at most n^(1+O(1/log log n)) unit distances.
- SupportedThe evidence favors the claim, but the examination is incomplete or the evidence is indirect.constitution →For arbitrarily large n there exist n-point planar sets with more than n^(1.014) unit-distance pairs.
- UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →There exist n-point planar sets where every subset has a distance repeated at least c|A|^2/n^(1-delta) times.
- ContradictedThe evidence, examined directly, weighs against the claim.constitution →The maximum number of unit distances among n points in the plane is at most n^(1+o(1)).
Where it is made
Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.
No source is recorded as making it.
Earlier ratings
- Sep 16, 2026
The inference is immediate once the premises are granted: a fixed exponent above 1 outgrows n^(1+O(1/log log n)) by an unbounded factor, so no reading of "asymptotically optimal" survives. The grid bound, that no scaling of the grid exceeds n^(1+O(1/log log n)) unit distances, is textbook and carries no risk; the argument therefore lives on the constructions, chiefly the refutation of the conjectured bound u(n) ≤ n^(1+o(1)), whose proof has been read whole and stands as contradicted, with Sawin's explicit exponent 1.014 and the Lee-Pohoata-Zhu construction each sufficient on its own. Either construction alone would carry the conclusion; both would have to fail for the argument to fail.