Geometric-progression-free sets over quadratic number fields

Abstract

In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. 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 geometricprogression-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.

Subsets of Fq[x] free of 3-term geometric progressions

“…It seems possible that a similar construction might yield a set with higher asymptotic density in F q [x]. In previous work, [Mc,Section 4], [BHMMPTW,Section 4.2] in different rings, improved upper bounds for the upper density were obtained by considering progressions among the smooth integers. Thus far this technique has not proven to be as useful in this ring, however it may just be that more work and computation are required.…”