Back to Search
Start Over
Geometric-progression-free sets over quadratic number fields
- Publication Year :
- 2014
- Publisher :
- arXiv, 2014.
-
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.<br />Comment: Corrected equations 4.4 and 4.5, other small changes, added a question about avoiding longer progressions
- Subjects :
- Discrete mathematics
Mathematics - Number Theory
General Mathematics
Unique factorization domain
Natural number
010103 numerical & computational mathematics
0102 computer and information sciences
Algebraic number field
01 natural sciences
Infimum and supremum
Ring of integers
Geometric progression
010201 computation theory & mathematics
FOS: Mathematics
Mathematics - Combinatorics
Number Theory (math.NT)
Combinatorics (math.CO)
0101 mathematics
Algebraic number
Dedekind zeta function
Mathematics
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....5708a32e317b4beecd7486223635ea1d
- Full Text :
- https://doi.org/10.48550/arxiv.1412.0999