  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.
