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## 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 ?
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Dear Adrien, I added the tag potential-theory since you and Margaret seem to think it would be helpful. – Hailong Dao Jul 11 2011 at 18:07
Nice initiative, thx! – Adrien Hardy Jul 12 2011 at 14:39

I do not have a satisfactory answer to your question, just a pointer. In the paper:

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.

BTW, the question could benefit from having an additional tag "Potential theory".

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Prof. Friedland, it looks nicer if you consolidate your posts, either through editing your answer or commenting on your answer. You also have enough reputation (at this writing) to leave comments to a question, if you prefer not to leave an answer. Regards. Gerhard "Email Me About System Design" Paseman, 2011.06.24 – Gerhard Paseman Jun 24 2011 at 23:08
Thank you for the pointer ! You're right about the additional tag. Nevertheless the tag "potential theory" does not exist yet and I don't have enough reputation points to create such a new tag ... – Adrien Hardy Jun 27 2011 at 14:13
@Gerhard: At your suggestion, my posts are now consolidated. @Adrien: I do not have enough reputation either; this comment was directed at "higher powers" in MO. – Margaret Friedland Jun 27 2011 at 15:38