Subsets of $\mathbb{F}_q[x]$ free of 3-term geometric progressions

TL;DR

This paper investigates the maximum density of subsets of the polynomial ring over finite fields that avoid 3-term geometric progressions, providing constructions and bounds that depend on the parameter q.

Contribution

It introduces a greedy construction for such subsets in _q[x] and analyzes how bounds vary with q, highlighting the impact of finite characteristic on the problem.

Findings

Constructed a greedy subset avoiding 3-term geometric progressions.

Derived bounds on the upper density of such subsets.

Showed bounds approach 1 as q increases.

Abstract

Several recent papers have considered the Ramsey-theoretic problem of how large a subset of integers can be without containing any 3-term geometric progressions. This problem has also recently been generalized to number fields, determining bounds on the greatest possible density of ideals avoiding geometric progressions. We study the analogous problem over $\mathbb{F}_q[x]$, first constructing a set greedily which avoids these progressions and calculating its density, and then considering bounds on the upper density of subsets of $\mathbb{F}_q[x]$ which avoid 3-term geometric progressions. This new setting gives us a parameter $q$ to vary and study how our bounds converge to 1 as it changes, and finite characteristic introduces some extra combinatorial structure that increases the tractibility of common questions in this area.