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An Exponent of 1.043 for the Unit Distance Problem

Primarily AI-generated textHuman understanding: some partsmath.CO — Combinatoricsmath.NT — Number Theory

Contributed by Eric Naslund

Version 2 / Oct 02, 2026 / CC BY 4.0

Abstract

Let u(U)u(U) count the unordered pairs at distance one in a finite planar set UU. We construct finite sets UjU_j with ∣Uj∣→∞|U_j|\to\infty and u(Uj)/∣Uj∣1.043171→∞u(U_j)/|U_j|^{1.043171}\to\infty. The largest exponent previously claimed, apart from preliminary versions of the present work, is 1.03581.0358, in the author's unpublished manuscript. The method is the number-field construction of OpenAI and Sawin. Here unit distances come from elements of relative norm one in quadratic extensions K/FK/F in which FF has both real and complex places, and the fields come from an infinite pro-22 tower over $\Q(\sqrt{241})$, ramified only above 22, 33 and 55, with Frobenius conditions at the two primes above 2929 and at one prime above 4141. The relative zeta value is bounded through the dihedral quotients of the tower group: the tower group has 2727 quotients isomorphic to the dihedral group of order~88; fifteen of them detect, by a census, the residue degree of the primes that split completely in the Kummer field up to norm 4⋅10134\cdot10^{13}, and seven of them generate a field EW′′E_{W''} of degree 81928192 contained in every field of the tower. We write ζEW′′\zeta_{E_{W''}} as a product of LL-functions of degrees 22, 44 and~88, evaluate the 576576 of degrees 44 and 88 rigorously at 2121 real points near~11, and combine the values through a kernel inequality with coefficients of both signs. Finite facts and numerical inequalities are certified by exact computation and interval arithmetic. Partial formalization PALOMAR-2026-10-01-000018 v2

Provenance statement

This work was done with extensive AI use, with agents developing the mathematics, writing the programs and writing the paper; I supervised, prompted, and coordinated them. The choice of the Frobenius condition at a prime above $41$, rather than at the prime $7$, was found in an agent run of the Muse Spark system. The refinements of the bound for the relative zeta value, the general shell weights, the certificates and this paper were produced with Claude Opus 5.5 working as an agent in Claude Code. Preliminary versions of this work, on which the construction builds, were produced with Claude Opus 5.5, ChatGPT Sol 5.6 and ChatGPT Astra 6. The reviews mentioned in Subsection 1.5 are automated reviews by AI agents in fresh contexts, not human peer review. I have had the agents provide a clearer introduction, and a dependency graph on page 4 to help navigate the paper and the proof, and make it easier to read. However I still recommend that the reader navigate this paper with their AI model of choice, and to ask the paper questions.
FormalizationsPalomar

Tools used

OpenAI
ChatGPTVersion Astra 6
OpenAI
ChatGPTVersion Sol 5.6
Anthropic
Claude OpusVersion 5.5
Meta
MuseVersion Spark 1.3 Contributor

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Version history

  1. v1Submitted by Eric NaslundInitial depositSupersededOct 01, 2026
  2. v2Submitted by Eric NaslundSubmitted as a new versionCurrentOct 10, 2026