Geometric Progression-Free Sequences with Small Gaps II

TL;DR

This paper investigates the existence of sequences of positive integers that avoid long geometric progressions while having small gaps, generalizing previous results to sequences where the gap size and the length of progressions grow.

Contribution

It extends prior work by establishing conditions under which $k$-GP-free sequences with small gaps exist for growing functions $h$ and $k$, broadening the understanding of geometric progression-free sequences.

Findings

Existence of $k$-GP-free sequences with small gaps under certain growth conditions.

Generalization from fixed to growing $k$ and $h$ functions.

Conditions relating $k(n)$ and $h(n)$ for such sequences to exist.

Abstract

When $k$ is a constant at least $3$, a sequence $S$ of positive integers is called $k$-GP-free if it contains no nontrivial $k$-term geometric progressions. Beiglb\"ok, Bergelson, Hindman and Strauss first studied the existence of a $ $$k$-GP-free sequence with bounded gaps. In a previous paper the author gave a partial answer to this question by constructing a $6$-GP-free sequence $S$ with gaps of size $O(\exp(6\log n/\log\log n))$. We generalize this problem to allow the gap function $k$ to grow to infinity, and ask: for which pairs of functions $(h,k)$ do there exist $k$-GP-free sequences with gaps of size $O(h)$? We show that whenever $(k(n)-3)\log h(n)\log\log h(n)\ge4\log2\cdot\log n$ and $h,k$ satisfy mild growth conditions, such a sequence exists.