  Let s and k be integers with s \geq 2 and k \geq 2. Let g_k^{(s)}(n) denote
the cardinality of the largest subset of the set {1,2,..., n} that contains no
geometric progression of length k whose common ratio is a power of s. Let
r_k(\ell) denote the cardinality of the largest subset of the set
{0,1,2,\ldots, \ell -1\} that contains no arithmetric progression of length k.
The limit \[ \lim_{n\rightarrow \infty} \frac{g_k^{(s)}(n)}{n} = (s-1)
\sum_{m=1}^{\infty}
  \left(\frac{1}{s} \right)^{\min \left(r_k^{-1}(m)\right)} \] exists and
converges to an irrational number.
