Breakthrough Tracker record
The minimal-distance problem has sharp exponent two-thirds
Pohoata constructs arbitrarily large point–line configurations whose off-diagonal distances are all at least n^(−2/3−ε). Combined with the published Cohen–Pohoata–Zakharov upper bound, this gives Δ_PL(n) = n^(−2/3+o(1)). The second version also derives finite-field matchings that disprove a related conjecture of Hunter, Pohoata, Verstraëte and Zhang.
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- Stable ID
math-minimal-distance-sharp-exponent-2026- Revision
math-minimal-distance-sharp-exponent-2026.v2- Field
- Mathematics · Discrete geometry and incidence theory
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional claimed solution
- Last checked
AI role
The author credits GPT-5.6 Sol Pro with manuscript assistance and with the decisive trace-zero-lattice idea that improved the construction to the sharp exponent. The author presents and takes responsibility for the proof.
Record details
- Problem or result
- Determine the asymptotic exponent of the greatest possible off-diagonal distance in an n-pair point–line incidence configuration in the unit square
- Authors
- Cosmin Pohoata
- Institutions
- Emory University
- Result date
- Preprint submitted July 22; second version posted July 30, 2026
Why it matters
The result determines the exponent of a discrete-geometry problem introduced in connection with the Heilbronn triangle problem. Its added finite-field consequence also overturns a conjectured upper bound for induced point–line matchings.
Limits
The matching lower bound is a 14-page second-version preprint without located independent specialist assessment or peer review. It determines the exponent, not a leading constant, and it does not settle the Heilbronn triangle problem.
Sources
- Primary: Cosmin Pohoata, The sharp exponent for the minimal distance problem
- Independent: Cohen, Pohoata and Zakharov, published upper bound in Inventiones Mathematicae
Correction and revision history
- 2026-07-23 — Added as a provisional first-version solution after checking the complete proof, the published matching upper bound, the exact asymptotic scope and the disclosed AI contribution.
- 2026-08-08 — Reviewed v2, added its finite-field consequence and removed the obsolete first-version caveat; status remains provisional.
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