Transfinite diameter of Bernstein sets in \(\mathbb{C}^N\) (Q701405)

A compact set \(K\) in \(\mathbb C^N\) is said to be a Bernstein set if there exists a constant \(M>0\) such that for every polynomial \(P\) and for every multiindex \(\alpha\in\mathbb N^N\) \[ \max\{|D^{\alpha}P(z)|: z\in K\}\leq M^{|\alpha|}(\deg P)^{|\alpha|}\max\{|P(z)|: z\in K\}. \] It follows from \textit{J. Siciak} [Zesz. Nauk. Uniw. Jagiell., Univ. Iagell. Acta Math. 1209(35), 37-45 (1997; Zbl 0951.32011)] that Bernstein sets are not pluripolar, whence they have positive transfinite diameter \(d(K)\) [see \textit{V. P. Zakharyuta}, Mat. Sb., N. Ser. 96 (138), 374-389 (1975; Zbl 0324.32009)]. It can be generalized as follows. A compact set \(K\subset\mathbb C^N\) is said to be a general Bernstein set if there exist a positive constant \(M\) and an infinite set \(T\subset\mathbb N\) such that for each \(\alpha\in\mathbb N^N\) with \(|\alpha|\in T\) and for each polynomial \(P\) of (total) degree \(|\alpha|\) one has \[ \max\{|D^\alpha P(z)|: z\in K\}\leq M^{|\alpha|}(\text{deg} P)^{|\alpha|}\max\{|P(z)|:\;z\in K\}. \] In this paper the authors show that if \(K\) is a general Bernstein set then \[ d(K)\geq\frac 1M\exp\left(-\sum_{k=1}^N\frac 1k\right). \] In particular, if \(K\) is a Bernstein set, then \(d(K)\geq 1/M2^{n-1}\). The Bernstein sets are special cases of Markov sets which are defined by the condition \[ \max\{|D^\alpha P(z)|: z\in K\}\leq M^{|\alpha|}(\deg P)^{r|\alpha|}\max\{|P(z)|: z\in K\} \] for each polynomial \(P\), with some positive constants \(M\) and \(r\) that are independent of \(P\) and \(\alpha\). By \textit{L. Białas-Cież} [Bull. Pol. Acad. Sci., Math. 46, No. 1, 83-89 (1998; Zbl 0978.41008)], if \(N=1\), then any Markov set is not polar. If \(N>1\), the question about non-pluripolarity of Markov sets remains still open.