Geometric Progression-Free Sequences with Small Gaps II

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.