 Main
Logarithmic capacity of Gdelta subsets of [0,1]
 Quintino, Fernando
 Advisor(s): Gorodetski, Anton
Abstract
We study the logarithmic capacity of Gdelta 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 Gdelta sets generated by such distribution satisfy our sufficient conditions almost surely. Hence, the random Gdelta sets have full capacity almost surely. This study is motivated by the Gdelta set of exceptional energies in the parametric version of the Furstenberg theorem on random matrix products. Additionally, we also study the family of Gdelta 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 redistribution construction can be considered as a generalization of a method applied by Ursell in his construction of a counterexample to a conjecture by Nevanlinna. Also, we propose a simple CauchySchwartz inequalitybased proof of related theorems by Lindeberg and by Erdos and Gillis.
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