We prove that the set of logarithms of moduli of entire functions of several complex variables is nowhere dense in the space of plurisubharmonic functions equipped with a topology that is a generalization of the topology of uniform convergence on compact sets. This topology is generated by a metric in which plurisubharmonic functions form a complete metric space. Thus, the logarithms of moduli of entire functions form a set of the first Baire category.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1602 – 1609, December, 2008.
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Girnyk, M.A. Logarithms of moduli of entire functions are nowhere dense in the space of plurisubharmonic functions. Ukr Math J 60, 1878–1888 (2008). https://doi.org/10.1007/s11253-009-0177-1
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DOI: https://doi.org/10.1007/s11253-009-0177-1