What is the growth rate in n of the denominator of the rational $r_n$, $n=1,2,3,\ldots$ in $1>r_n>1-e^{-n}$ with the smallest denominator?
If $I_n$ is a (sufficiently random) sequence of disjoint intervals in R>0, whose lenght (ie elementary measure) is given by a f(n) decreasing to 0, with $\sum_n f(n) = \infty$, what is the expected growth rate of the lowest denominator of all rationals in $I_n$ ?