Phase transition of capacity for the uniform $G_{\delta}$-sets

TL;DR

This paper investigates the capacity of certain dense $G_{\delta}$ sets in [0,1], revealing a phase transition from full to zero capacity as the intervals' lengths decrease, with implications for exceptional energies in random matrix theory.

Contribution

It introduces a new analysis of capacity phase transitions in $G_{\delta}$ sets and generalizes a redistribution method, connecting to classical results in complex analysis and probability.

Findings

Identifies a sharp phase transition in capacity based on interval decay rate.

Provides a generalized redistribution construction method.

Offers a simple proof of classical theorems using Cauchy-Schwartz inequality.

Abstract

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.