Geometricprogressionfree sets over quadratic number fields
Abstract
A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct highdensity subsets of the algebraic integers of an imaginary quadratic number field that avoid 3term 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 geometricprogressionfree sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometricprogressionfree 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2014
 DOI:
 10.48550/arXiv.1412.0999
 arXiv:
 arXiv:1412.0999
 Bibcode:
 2014arXiv1412.0999B
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Combinatorics
 EPrint:
 Corrected equations 4.4 and 4.5, other small changes, added a question about avoiding longer progressions