The sharp exponent for the minimal distance problem

2026/07/22 by Cosmin Pohoata
Mathematics · #math.CO #math.MG #math.NT

paper · pdf

14 pages, new Section 5 added

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

We show that for every fixed ε>0, there exist arbitrarily large families of point-line pairs (x₁,ℓ₁),…,(x_n,ℓ_n) in [0,1]², with x_i ∈ ℓ_i for all i, and such that dist(x_i,ℓ_j)≥ n^-2/3-ε for all i ≠ j. Combined with a previous result of Cohen, the author, and Zakharov, this solves the minimal distance problem. The same construction also comes with an unexpected finite field consequence: for every ε>0, there exists a set of primes q of positive relative density for which \mathbb F_q² contains an induced point-line matching of size \gtrsim q^3/2-ε. This disproves a conjecture of Hunter, the author, Verstraëte and Zhang.

Citations

• The sum-product conjecture is false for real numbers
• Remarks on the disproof of the unit distance conjecture
• A new upper bound for sets with no square differences

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• Nearly equal distances in the plane, II
• An explicit lower bound for the unit distance problem
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