# Compact sets of the complex plane having the K-property ?

I would like to have a better understanding of a notion I've met in the beautiful book of Nikishin and Sorokin "Rational 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 $D(z,\rho)$ 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 ?
• Dear Adrien, I added the tag potential-theory since you and Margaret seem to think it would be helpful. Jul 11 '11 at 18:07
• Nice initiative, thx! Jul 12 '11 at 14:39

Białas-Cież, Leokadia Markov sets in ${\bf C}$ are not polar. Bull. Polish Acad. Sci. Math. 46 (1998), no. 1, 83–89
for a compact subset $E$ of $\mathbb{C}$ which satisfies Markov inequality (i.e., certain estimate for derivatives of polynomials) a lower bound is proved for the capacity of $E$ in terms of the diameter of $E$ raised to the power 1/3. This is a global estimate, not a local one that you are asking about, but there may be a relation. The author of the paper may know more, so I suggest asking her anyway, whether you get any more specific answers on MO or not.