  We consider a family of dense $G_{\delta}$ subsets of $[0,1]$, defined as
intersections of unions of small uniformly distributed intervals, and study
their capacity. Changing the speed at which the lengths of generating intervals
decrease, we observe a sharp phase transition from full to zero capacity. Such
a $G_{\delta}$ set can be considered as a toy model for the set of exceptional
energies in the parametric version of the Furstenberg theorem on random matrix
products.
  Our re-distribution construction can be considered as a generalization of a
method applied by Ursell in his construction of a counter-example to a
conjecture by Nevanlinna. Also, we propose a simple Cauchy-Schwartz
inequality-based proof of related theorems by Lindeberg and by Erd\"os and
Gillis.
