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Let us consider an entire function of several complex variables $f(z_1,\dots,z_n)$ and its modulus maximum $$M(r,f):=\max \{ |f(z_1,\dots,z_n)|: |z_1|\le r,\dots,|z_n| \le r \} $$ with $r\ge 0$. How big may the set $\{(z_1,\dots,z_n):|f(z_1,\dots,z_n)|=M(r,f), |z_1|=r,\dots,|z_n|=r\} $ (It is well known that the maximum is attained on the skeleton of the polycylinder.) be?

In the one-dimensional case the answer is known. Let $\nu(r; g)$ denote the number of maximum modulus points of an entire function $f(z)$ on the circle $|z|=r$.In 1964 Erdos set up the question whether it is possible to find an entire function $f(z)\neq c z^n$ with $\nu(r; f)$ unbounded. In 1968 F.Herzog and G.Piranian [The counting function for points of maximum modulus. In:Proc.Symp.PureMath.(1968),v.11.Entire Functions and Related Parts Analysis, AMS, p. 240-243 ] gave a positive answer to this question.They constructed an entire function $f(z)$ of infinite order with $\nu(r, f) \to \infty$ as $r \to \infty$. E. Ciechanowicz and I. Marchenko [A note on the separated maximum modulus points of meromorphic functions. (English) Zbl 1295.30069 Ann. Pol. Math. 110, No. 3, 295-310 (2014) ] proved the sharp upper estimate of the number of separated maximum modulus points for meromorphic functions of finite lower order.

However, the case of higher dimensions substantionally differs from the one-dimesional case: the maximum set is infinite. In view of it is it possible to estimate this set in terms of some measure?

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