Phase Transition of Logarithmic Capacity for the Uniform G?-Sets

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