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arXiv · 1910.07653

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

Victor Kleptsyn·Fernando Quintino
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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.

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Victor Kleptsyn, Fernando Quintino. 2019-10-16. Phase transition of capacity for the uniform $G_{\delta}$-sets. https://doi.org/10.1007/s11118-020-09896-8

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