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 logarithmic 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.

我们考虑一个家庭密集的ģ _δ [0，1]的子集，定义为小均匀分布间隔工会交点，并研究其对数的能力。通过改变生成间隔长度减小的速度，我们观察到从满容量到零容量的急剧相变。这种ģ _δ集可以被认为是用于在弗斯滕伯格定理的随机矩阵的产品的参数版本的组特殊的能量的玩具模型。我们的重新分配构造可以看作是Ursell在其对Nevanlinna的猜想的反例构造中采用的方法的概括。此外，我们提出了一个简单的基于Cauchy-Schwartz不等式的相关定理的证明，由Lindeberg以及Erdös和Gillis提出。