Phase transition of capacity for the uniform $G_δ$-sets

TL;DR

This work analyzes the logarithmic capacity of uniform $G_δ$-sets $S$ on $[0,1]$ formed as intersections of unions of small evenly distributed intervals. By introducing a redistribution technique, it shows a sharp phase transition in capacity as the interval length $r_n$ decays: subexponential decay preserves full capacity, while exponential decay with $r_n=e^{-n^ alpha}$ yields zero capacity for $ alpha>2$ and full capacity for $ alpha<2$, with the borderline case $ alpha=2$ likely still full capacity. The results connect to exceptional energies in parametric Furstenberg theorems and generalize Erdős–Gillis–Lindeberg-type volume criteria, using CS-based proofs and multi-level redistribution arguments. The paper also presents a counterexample demonstrating non-continuity of capacity on bounded intervals, underscoring subtlety in capacity behavior beyond standard measures. Overall, the work provides a robust framework for understanding how fine-grained interval constructions influence capacity and their relevance to dynamical systems with random matrix products.

Abstract

We consider a family of dense $G_δ$ 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 $G_δ$ 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 Erdös and Gillis.

Figures (8)

• Figure 1: Sets $V_n$
• Figure 2: The idea of a re-distribution
• Figure 3: Self-interaction (dark squares) and outer-interaction (light ones) parts of the energy integral for the re-distributed measure $R(\mu|V_n)$.
• Figure 4: Transforming the density $f(x)$ into a continuous one.
• Figure 5: Two intervals $J,J'$ and their centers.

Theorems & Definitions (43)

• Definition
• Theorem 1.1: Subexponential uniform $G_\delta$
• Remark
• Theorem 1.2: Phase transition
• Theorem 1.3: P. Erdös, J. Gillis, E-G, generalizing Lindeberg Lin
• Corollary 1.4
• Conjecture : Nev; see also E-G
• Theorem 1.5
• Proposition 1.6
• Corollary 1.7