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I would like to have a better understanding of a notion I've met in the beautiful book of Nikishin and Sorokin "Rational approximations approximation and Orthogonality", since they do not provide examples.

As it is classical to do in potential theory, denote for $\mu$ in $M_1(K)$, the set of probability measures on a compact set $K\subset \mathbb{C}$, its logarithmic energy by $$I(\mu)=\iint \log\frac{1}{|x-y|}d\mu(x)d\mu(y)$$ and define the capacity of a compact set $K\subset\mathbb{C}$ as $$Cap(K)=\exp\Big(-\inf_{\mu\in M_1(K)} I(\mu)\Big).$$ $K$ is said to satisfy the K-property at $z\in K$ if there exists $\rho_z > 0$ and $k_z> 0$ such that $$Cap(K\cap D(z,\rho))\geq \rho^{k_z}$$ for any $0< \rho < \rho_z$, where $D(z,\rho)$ stand stands for the disc centered at $z$ with radius $\rho$. We say that $K$ satisfies the K-property if it satisfies the K-property at every $z\in K$.

One can show that segments, or circles, satisfy this K-property.

Questions :

• Example of compact sets with positive capacity which do not have the K-property ?
• More generally, do you have references about K-property for compact sets ?
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I would like to have a better understanding of a notion I've met in the beautiful book of Nikishin and Sorokin "Rational approximations and Orthogonality", since they do not provide examples.

As it is classical to do in potential theory, denote for $\mu$ in $M_1(K)$, the set of probability measures on a compact set $K\subset \mathbb{C}$, its logarithmic energy by $$I(\mu)=\iint \log\frac{1}{|x-y|}d\mu(x)d\mu(y)$$ and define the capacity of a compact set $K\subset\mathbb{C}$ as $$Cap(K)=\exp\Big(-\inf_{\mu\in M_1(K)} I(\mu)\Big).$$ $K$ is said to satisfy the K-property at $z\in K$ if there exists $\rho_z$ \rho_z > 0$and$k_z$k_z> 0$ such that $$Cap(K\cap D(z,\rho))\geq \rho^{k_z}$$ for any $0< \rho < \rho_z$, where $D(z,\rho)$ stand for the disc centered at $z$ with radius $\rho$. We say that $K$ satisfies the K-property if it satisfies the K-property at every $z\in K$.

One can show that segments, or circles, satisfy this K-property.

Questions :

• Example of compact sets with positive capacity which do not have the K-property ?
• More generally, do you have references about K-property for compact sets ?
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