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The Minimal Distance Problem

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minimal-distance-problem-sharp-exponentGeometry & topologyposed by Cohen, Pohoata, Zakharovrecorded: solved

1 attempt · no person has looked

Statement

How well separated can a family of point-line pairs in the unit square be? For every ε>0\varepsilon > 0 there are arbitrarily large families (x1,ℓ1),…,(xn,ℓn)(x_1,\ell_1),\ldots,(x_n,\ell_n) in [0,1]2[0,1]^2 with xi∈ℓix_i \in \ell_i and dist(xi,ℓj)≥n−2/3−ε\mathrm{dist}(x_i,\ell_j) \ge n^{-2/3-\varepsilon} for all i≠ji \ne j. Combined with earlier work of Cohen, Pohoata and Zakharov this settles the problem at the sharp exponent 2/32/3. The same construction disproves a conjecture of Hunter, Pohoata, Verstraete and Zhang about induced point-line matchings over finite fields.

Context

also disproves a separate finite-field conjecture of Hunter, Pohoata, Verstraete and Zhang

A named problem in incidence geometry with a documented ladder of exponents and connections to square-difference-free sets, Nikodym sets and minimal blocking sets.

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  • #1

    Attempt 1

    constructionGPT-5.6 Pro with Cosmin Pohoata ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5.6 Pro
    people
    Cosmin Pohoata

    The acknowledgement draws the line precisely. The author's own plan was to use a high-degree number field analogue of the Hunter-Pohoata-Verstraete-Zhang construction to reach the Ruzsa endpoint. In his words, the decisive new idea of using the codimension-one, square-difference-free, trace-zero lattice in place of a Ruzsa-like set, which is what upgrades the exponent to the sharp one, is entirely due to GPT-5.6 Pro.

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