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I would like to know some examples of the following polar sets (if they exist):

  • a non trivial uncountable polar set in $\mathbb{R}^{2}$;
  • a polar set in $\mathbb{R}^{2}$ contained in $\mathbb{R}$ with positive Lebesgue measure over $\mathbb{R}$.

I recall that a set $E\subset \mathbb{R}^{2}$ is called polar if there exists a subharmonic function $u:\mathbb{R}^{2}\to \mathbb{R}\cup \{-\infty\}$ such that $E\subset \{u=-\infty\}$.

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  • $\begingroup$ A Cantor set will do the job. Take a fat one if you like positive $1$-D Lebesgue measure. In general, any set with positive Hausdorff dimension will be non-polar. $\endgroup$ – Mateusz Kwaśnicki Oct 15 '19 at 10:33
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    $\begingroup$ Thanks. Anyway, can you be more precise or give me some references? Because a Cantor set is not necessarily polar. I tried to estimate the capacity of a general Cantor set using its construction, but in my estimate when I add the condition to be polar, the Lebesgue measure becomes 0. $\endgroup$ – Trusio Oct 15 '19 at 13:31
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    $\begingroup$ Whoops, I misread your question! Of course Cantor sets will typically be non-polar, sorry. I suppose your second question made me think that you are asking about non-polar sets, because as explained in Josiah Park's answer, the logarithmic potential of the 1-D Lebesgue measure on $E \subset \mathbb{R} \times \{0\}$ is bounded on $E$, so if $E$ has positive Lebesgue measure, it is non-polar. $\endgroup$ – Mateusz Kwaśnicki Oct 15 '19 at 18:49
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There is a construction mentioned in Ransford (Potential Theory in the Complex Plane, pg 143) which supplies an uncountable polar set in $\mathbb{C}$. It is reproduced below.

Letting $\{s_n\}_{n}$ be a sequence of positive numbers, $s_n<1$ and $C(s_1,\dots,s_n)$ be the set obtained by removing the middle of each interval in $C(s_1,\dots,s_{n-1})$ an open subinterval of length proportion $s_n$ of the interval (and so on, starting initially with the interval $[0,1]$), finally set

$$C(s)=\cap_{n\geq 1} C(s_1,\dots,s_n).$$

When $s_n=1-(1/2)^{2^n}$, then $C(s)$ is polar.

As mentioned in the comments, general Cantor sets will not work, and it is an open question to determine the capacity of the standard Cantor set, which is estimated $c(E) \sim 0.220949102189507$.

There is not a set which satisfies the second hypothesis. Any subset of the real line of positive Lebesgue measure has positive capacity and so is not polar (see Theorem 5.3.2 in Ransford).

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A polar set of positive measure on a line does not exist. Every polar set has zero capacity, and thus zero $1$-measure, and zero Hausdorff dimension. An uncountable polar set can be constructed as a Cantor type set but not the standard one, the size of the segments of $n$-th generation has to decrease very fast (faster than exponentially).

Ref. Nevanlinna, Analytis functions, Ch. V, section 6, or L. Carleson, Selected problems on exceptional sets, Van Nostrand Co., Inc., Princeton, 1967.

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