An explicit lower bound for the unit distance problem
Will Sawin has presented a new explicit lower bound for the unit distance problem in a recent paper. This work surpasses previous findings by providing a clearer exponent and disproving a long-standing conjecture by Erdős. The methodology involves advanced number-theoretic techniques to construct specific algebraic structures.
- ▪The paper shows that there are sets of n points in the plane with n arbitrarily large that contain more than n^{1.014} pairs of points separated by a distance exactly 1.
- ▪This result improves upon recent work by a team at OpenAI, which had an inexplicit exponent greater than 1.
- ▪The approach relies on constructing algebraic number fields of large degree and small discriminant using a Golod-Shafarevich criterion.
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Mathematics > Combinatorics arXiv:2605.20579 (math) [Submitted on 20 May 2026] Title:An explicit lower bound for the unit distance problem Authors:Will Sawin View a PDF of the paper titled An explicit lower bound for the unit distance problem, by Will Sawin View PDF HTML (experimental) Abstract:We show that there are sets of $n$ points in the plane with $n$ arbitrarily large that contain more than $n^{1.014}$ pairs of points separated by a distance exactly $1$. This improves on very recent work of a team at OpenAI, who proved the same result with an inexplicit exponent greater than $1$, drastically improving on the best previous lower bound and disproving a conjecture of Erdős.
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