An explicit lower bound for the unit distance problem

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. The method is number-theoretic, relying on constructing algebraic number fields of large degree and small discriminant with many primes of small norm via a Golod-Shafarevich criterion argument.

• The paper presents an explicit construction using algebraic number theory to achieve a lower bound of n^1.014 for unit distances in planar sets.
• It leverages CM fields, lattice projections, and explicit class group bounds to optimize the density of unit distance pairs.
• The methodology refutes Erdős's linear conjecture and establishes quantitative benchmarks for future research in discrete geometry and algebraic methods.

Explicit Lower Bounds for the Unit Distance Problem in the Plane

Introduction

This paper addresses the long-standing unit distance problem in discrete geometry, focusing on explicit lower bounds for the maximum number of pairs of points at unit distance in subsets of the Euclidean plane with $n$ points. The approach employs advanced tools from algebraic number theory, offering a significant improvement over previous results by providing both an explicit exponent and a construction for large sets exhibiting many unit distances. The work materially extends and makes quantitative the recent breakthrough of the OpenAI team, strengthening the refutation of a classical conjecture of Erdős.

Background and Previous Bounds

The unit distance problem, as posed by Erdős, asks for the maximal number of pairs of points at unit distance that a planar $n$-point set can possess. The best-known upper bound is $O(n^{4/3})$, due to Spencer, Szemerédi, and Trotter. Until very recently, the most effective lower bound, given by Erdős, took the form $n^{1 + c/\log\log n}$ for some $c > 0$. Erdős conjectured that the true growth was $n^{1 + o(1)}$, suggesting essentially linear behavior.

This conjecture has now been refuted. Both an OpenAI model-assisted proof and an independent, human-crafted exposition established the existence of sets achieving more than $n^{1+\delta}$ unit distances for some $\delta > 0$, though with non-explicit or minuscule exponents (e.g., $\delta \approx 6\times10^{-38}$ in the latter case).

Main Contributions

The principal result in this paper is the explicit construction, for arbitrarily large $n$, of planar sets of size $n$0 containing at least $n$1 pairs of points at unit distance, up to an absolute constant. The approach makes every parameter in the construction explicit, thereby substantially increasing the achieved exponent compared to prior explicit work. This confirms for the first time the existence of large sets with a polynomial improvement ($n$2) over the linear lower bound in the unit distance context.

Methodology

The construction harnesses algebraic number fields—specifically, CM fields with totally real subfields—and their rich arithmetic to design high-dimensional lattices admitting many short vectors whose projections to $n$3 all have length one. By refining arguments based on properties of ideals and class groups in these number fields, the density of such unit vectors is optimized. The paper relies on the following core elements:

• Lattice Construction via Number Fields: Integer rings and their ideals in number fields are projected to the plane. CM fields and their totally real subfields allow projections with controlled norm.
• Unit Distance Realization: The number of pairs at unit distance corresponds to the number of vectors in the lattice with prescribed norms.
• Explicit Class Group Bounds: The Louboutin bound for relative class numbers is used instead of the classical Minkowski bound, yielding stronger estimates.
• Golod-Shafarevich Criterion: A criterion from Galois cohomology is manipulated to ensure the existence of infinite towers of number fields with small relative discriminants and many split primes. By controlling local splitting and inertia degrees using modified class field towers, the approach guarantees a highly structured supply of suitable ideals.

The proofs are fully explicit—each step is quantitatively stated, and all parameters (e.g., the exponent $n$4) are computed concretely, culminating in the main lower bound.

Strong Numerical and Contradictory Claims

Theorem: There exist planar point sets of size $n$5 containing more than $n$6 pairs at unit distance, for an absolute constant $n$7 and arbitrary large $n$8.

This assertion not only improves known constructive lower bounds but also directly contradicts the Erdős conjecture that this growth is at most $n$9. Furthermore, an explicit upper bound is derived: the methodology cannot yield exponents exceeding $O(n^{4/3})$0, i.e., $O(n^{4/3})$1, even in the most optimistic scenario subject to existing techniques.

Technical and Practical Implications

Theoretically, this result demonstrates that interactions between algebraic number theory and discrete geometry can unravel and unsettle classical combinatorial conjectures. The use of explicit towers of number fields with prescribed ramification and splitting properties constitutes a powerful tool, likely extensible to other geometric extremal problems where coordinate algebraic structure is advantageous.

Practically, the result does not suggest improved constructions for concrete, finite $O(n^{4/3})$2—the constants involved and the necessity of large degree number fields make immediate geometric applications limited. However, the explicit quantitative nature of the bounds provides essential stepping stones for computational advances and motivates further investigations into the effective list decoding of algebraic geometric codes and higher-dimensional analogues.

Future Directions

Several avenues warrant exploration:

• Analytic Number Theory Enhancements: Improvements in lower bounds for the relative class number could directly sharpen the exponent $O(n^{4/3})$3.
• Computational Search: Systematic, possibly ML-driven search for sets $O(n^{4/3})$4, $O(n^{4/3})$5, and parameter optimizations may drive further advances.
• Broader Applicability: The interplay between the splitting of primes and lattice packing suggests potential for similar constructions in problems of sphere packings, kissing numbers, and distance set cardinalities in higher dimensions.

Conclusion

This paper establishes, via explicit and constructive number-theoretic methods, that planar sets with superlinear numbers of unit distances exist for all sufficiently large $O(n^{4/3})$6, with an explicit exponent ($O(n^{4/3})$7). This advances and codifies recent progress, providing a detailed technical framework that is likely to inspire further cross-pollination between combinatorics, number theory, and algebraic geometry in geometric extremal problems. The explicitness of the bounds and the refined arithmetic analysis set a new standard for quantitative results in the field.

Reference: "An explicit lower bound for the unit distance problem" (2605.20579)