Admissible sets of integers in imaginary quadratic fields of class number 1

Abstract

Let $\mathcal O_K$ be the ring of integers of a number field $K$. A finite subset $\mathcal H=\{η_1,\ldots,η_k\}\subset\mathcal O_K$ is said to be \emph{admissible} if for every prime ideal $\mathfrak p$ in $\mathcal O_K$, the reductions of the $η_i$ modulo $\mathfrak p$ do not represent every residue class. The diameter of a finite subset $S$ of $\mathcal O_K$ is defined to be \[\mathrm{diam}(S):= \max_{η_i,η_j\in S:η_i\neqη_j}|\mathcal N(η_i-η_j)|\] where $\mathcal N$ 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.