Geometric Progression-Free Sequences with Small Gaps

Abstract

Various authors, including McNew, Nathanson and O'Bryant, have recently studied the maximal asymptotic density of a geometric progression free sequence of positive integers. In this paper we prove the existence of geometric progression free sequences with small gaps, partially answering a question posed originally by Beiglb\"ock et al. Using probabilistic methods we prove the existence of a sequence $T$ not containing any $6$-term geometric progressions such that for any $x\geq1$ and $\varepsilon>0$ the interval $[x,x+C_{\varepsilon}\exp((C+\varepsilon)\log x/\log\log x)]$ contains an element of $T$, where $C=\frac{5}{6}\log2$ and $C_{\varepsilon}>0$ is a constant depending on $\varepsilon$. As an intermediate result we prove a bound on sums of functions of the form $f(n)=\exp(-d_{k}(n))$ in very short intervals, where $d_{k}(n)$ is the number of positive $k$-th powers dividing $n$, using methods similar to those that Filaseta and Trifonov used to prove bounds on the gaps between $k$-th power free integers.