Geometric-progression-free sets over quadratic number fields

TL;DR

This paper extends the study of large subsets avoiding 3-term geometric progressions from natural numbers to quadratic number fields, constructing high-density examples and establishing bounds using algebraic number theory tools.

Contribution

It introduces methods to construct and analyze geometric-progression-free sets over quadratic number fields, generalizing previous rational integer results and providing density bounds via Dedekind zeta functions.

Findings

Constructed high-density geometric-progression-free subsets in imaginary quadratic fields.

Derived upper bounds for the density of such sets using generalized Riddell arguments.

Established lower bounds for the supremum of upper densities of these sets.

Abstract

A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets "greedily," a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.