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Logarithmic capacity of G-delta subsets of [0,1]


We study the logarithmic capacity of G-delta subsets of the interval [0,1]. Let {J_k} be a countable collection of open intervals with centers {c_k} in (0,1) and with lengths {l_k} that decrease to 0 as k approaches infinity. Let S be the collection of points that belong to infinitely many J_k. We provide sufficient conditions for S to have full capacity, i.e. Ca(S)=Ca([0,1]). We consider the case when the intervals decay exponentially and are placed in [0,1] randomly with respect to some given distribution. We prove that the random G-delta sets generated by such distribution satisfy our sufficient conditions almost surely. Hence, the random G-delta sets have full capacity almost surely. This study is motivated by the G-delta set of exceptional energies in the parametric version of the Furstenberg theorem on random matrix products. Additionally, we also study the family of G-delta sets {S(alpha)} with alpha>0 that are generated by setting the intervals to l_k=e^{-k^\alpha}. We observe a sharp transition at alpha=1 from full capacity to zero capacity by varying \alpha>0.

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 Erdos and Gillis.

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