  We consider distributional approximation by generalized Dickman
distributions, which appear in number theory, perpetuities, logarithmic
combinatorial structures and many other areas. We prove bounds in the
Kolmogorov distance for the approximation of certain weighted sums of Bernoulli
and Poisson random variables by members of this family. While such results have
previously been shown in Bhattacharjee and Goldstein (2019) for distances based
on smoother test functions and for a special case of the random variables
considered in this paper, results in the Kolmogorov distance are new. We also
establish optimality of our rates of convergence by deriving lower bounds. As a
result, some interesting phase transitions emerge depending on the choice of
the underlying parameters. The proofs of our results mainly rely on the use of
Stein's method. In particular, we study the solutions of the Stein equation
corresponding to the test functions associated to the Kolmogorov distance, and
establish their smoothness properties. As applications, we study the runtime of
the Quickselect algorithm and the weighted depth in randomly grown simple
increasing trees.
