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A geometric progression of length $k$ and integer ratio is a set of numbersof the form $\{a,ar,\dots,ar^{k-1}\}$ for some positive real number $a$ andinteger $r\geq 2$. For each integer $k \geq 3$, a greedy algorithm is used toconstruct a strictly decreasing sequence $(a_i)_{i=1}^{\infty}$ of positivereal numbers with $a_1 = 1$ such that the set \[ G^{(k)} =\bigcup_{i=1}^{\infty} \left(a_{2i} , a_{2i-1} \right] \] contains no geometricprogression of length $k$ and integer ratio. Moreover, $G^{(k)}$ is a maximalsubset of $(0,1]$ that contains no geometric progression of length $k$ andinteger ratio. It is also proved that there is a strictly increasing sequence$(A_i)_{i=1}^{\infty}$ of positive integers with $A_1 = 1$ such that $a_i =1/A_i$ for all $i = 1,2,3,\ldots$.  The set $G^{(k)}$ gives a new lower bound for the maximum cardinality of asubset of the set of integers $\{1,2,\dots,n\}$ that contains no geometricprogression of length $k$ and integer ratio.

A problem of Rankin on sets without geometric progressions