Polynomial Unit-Distance Lower Bounds over Prime Fields
Manuscript 5 October 2026 · Online 5 October 2026
Abstract
For each \(a\in\{1,3\}\), infinitely many primes \(p\equiv a\pmod4\) admit a set of exactly \(p\) points in \(\mathbb F_p^2\) with at least \(p^{1.0058}\) unordered pairs at quadratic distance one.
We transfer Sawin's number-field unit-distance construction to prime fields using a field-uniform quantitative Chebotarev estimate of Zaman. A relative ideal-norm bound prevents collisions without a bounded global denominator. Split and inert reductions give the required quadratic form, and disjoint translates supply exactly \(p\) points. Rational interval arithmetic certifies the exponent.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Scoped lower-bound theorem
- Categories
- math.CO · math.NT
- Manuscript
- 5 October 2026
- Online release
- 5 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “Polynomial Unit-Distance Lower Bounds over Prime Fields,” EulerSolve Research Papers, AIM-COMBINATORICS-0257, 2026. https://doi.org/10.5281/zenodo.23169763.
BibTeX
@misc{Ferudun2026PrimeUnitDistances,
author = {Ferudun, Alper},
title = {Polynomial Unit-Distance Lower Bounds over Prime Fields},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/aim-combinatorics-0257/},
doi = {10.5281/zenodo.23169763},
note = {AIM-COMBINATORICS-0257; unrefereed preprint}
}