Admissible sets of integers in imaginary quadratic fields of class number 1
Abstract
Let 𝒪K be the ring of integers of a number field K. A finite subset ℋ={η1,…,ηk}⊂𝒪K is said to be \emph{admissible} if for every prime ideal p in 𝒪K, the reductions of the ηi modulo p do not represent every residue class. The diameter of a finite subset S of 𝒪K is defined to be diam(S):= maxηi,ηj∈ S:ηi≠ηj|𝒩(ηi−ηj)| where 𝒩 denotes the norm down to the rational field. This paper computationally investigates admissible subsets of rings of integers in imaginary quadratic fields of class number 1 (such as the Gaussian integers or the Eisenstein integers) that are large relative to their diameter. The motivation is twofold. Firstly, there is the improvement of results analogous to the question of an upper bound for the least gap between large rational primes, which has been one of the great number theoretic successes of the twenty-first century. A second motivation is that there is the opportunity to disprove a variation of the second Hardy-Littlewood conjecture. The calculations in the paper provide new records on the smallest distance between primes in the fields in question, and explicit counter-examples to a generalisation of the second Hardy-Littlewood conjecture in all bar one of the fields.
Related
- Imaginary quadratic fields of class number at most 2000
- Maximal unramified 3-extensions of imaginary quadratic fields and \textrm SL\sb 2(\mathbb Z\sb 3)
- p-class groups in the cyclotomic ℤp-extension of imaginary quadratic fields
- Automorphic symbols and automorphic \emph{L}-values of GL(2) over imaginary quadratic fields
- The enhanced polylogarithm conjecture in weight two for imaginary quadratic fields
BibTeX
@misc{shiu2026admissible,
title = {{Admissible sets of integers in imaginary quadratic fields of class number 1}},
author = {Daniel Shiu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34556},
eprint = {2610.34556},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34556},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Admissible sets of integers in imaginary quadratic fields of class number 1
AU - Daniel Shiu
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34556
UR - https://arxiv.org/abs/2610.34556
AB - Let 𝒪_K be the ring of integers of a number field K. A finite subset ℋ={η_1,…,η_k}⊂𝒪_K is said to be \emph{admissible} if for every prime ideal p in 𝒪_K, the reductions of the η_i modulo p do not represent every residue class. The diameter of a finite subset S of 𝒪_K is defined to be diam(S):= max_η_i,η_j∈ S:η_i≠η_j|𝒩(η_i−η_j)| where 𝒩 denotes the norm down to the rational field. This paper computationally investigates admissible subsets of rings of integers in imaginary quadratic fields of class number 1 (such as the Gaussian integers or the Eisenstein integers) that are large relative to their diameter. The motivation is twofold. Firstly, there is the improvement of results analogous to the question of an upper bound for the least gap between large rational primes, which has been one of the great number theoretic successes of the twenty-first century. A second motivation is that there is the opportunity to disprove a variation of the second Hardy-Littlewood conjecture. The calculations in the paper provide new records on the smallest distance between primes in the fields in question, and explicit counter-examples to a generalisation of the second Hardy-Littlewood conjecture in all bar one of the fields.
ER -