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The sharp exponent for the minimal distance problem

Cosmin Pohoata

Pohoata constructs n incident point-line pairs in [0,1]² whose off-diagonal distances are at least n^(−2/3−ε) for every ε>0 and all sufficiently large n. With the previous upper bound, the maximum separation is n^(−2/3+o(1)).

Sources for this record

What was already known

Cohen, Pohoata and Zakharov established the n^(−2/3+o(1)) upper bound. Earlier constructions gave a lower bound near n^(−0.810759); Pohoata’s original number-field route targeted exponent 4/5.

AI involvement AI-centered

Pohoata credits GPT-5.6 Pro with the decisive trace-zero, square-difference-free lattice. He supplied the number-field plan and developed the counting and Euclidean realization.

The model supplied the decisive retained construction; Pohoata supplied the research direction and completed exposition and argument.

Importance Major

Closes the exponent gap in the point-line minimal-distance problem.

Limitations

The result is a sharp asymptotic exponent, not a fixed endpoint constant. The manuscript contains a proof, but no separate completed proof check is documented.

Technical verification Reported

The paper gives a detailed proof but does not document a separate completed proof check or formalization. Reported is the evidence default.

Human understanding Full

Pohoata explains that trace positivity forbids square differences and field norms provide quantitative Euclidean separation; his proof develops this mechanism.

Correction and publication history