Suppose we have the following problem:
We start with an interval of length $1$ and break it into two intervals of lengths $r$ and $1-r$, where $r$ is a random variable in $[0,1]$ with probability density function $\eta(r)$. For example choose $\eta(r)=1$ and we break the interval at a uniform random position.
Now any interval that is remaining and is larger than $x_c$, a fixed threshold $0<x_c<1$, again breaks according to the same rule that is if the current length of the interval is $x_i>x_c$ it will break into $x_i r$ and $x_i(1-r)$. All the $r$'s are independent of each other but with the same $\eta(r)$.
Continue until no interval of length $x>x_c$ exists.
What is the final average number of fragments with size less than $x$, denoted $C(x)$. Clearly $C(x)$ for $x>x_c$ is equal to $C(x_c)$ so it is sufficient to find $C(x)$ for $0\leq x\leq x_c$.
I asked myself this question while reading "Phase transition in a random fragmentation problem with applications to computer science" where the authors look at a generalised fragmentation process of breaking an interval into $m$ pieces and analyse the average number of interval splits.