Geometric progression-free sequences with small gaps. II.

zbMATH Open1414.11015arXiv1503.06906MaRDI QIDQ2830377

Xiaoyu He

Publication date: 28 October 2016

Published in: Integers (Search for Journal in Brave)

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 ${\displaystyle }$$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 $O(exp(6logn/loglogn))$. 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 $(k(n)-3)logh(n)loglogh(n)ge4log2cdotlogn$ and $h,k$ satisfy mild growth conditions, such a sequence exists.

Full work available at URL: https://arxiv.org/abs/1503.06906

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Mathematics Subject Classification ID

Density, gaps, topology (11B05) Distribution of integers with specified multiplicative constraints (11N25)

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