Entire functions of exponential type,increasing slowly along the real hyperplane

Entire functionsf(z), zC ⁿ , of exponential type at most a and bounded on subsets E of the real hyperplane, are investigated. It is known that if E is relatively dense with respect to the Lebesgue measure or it is an -net inR ⁿ , then such f(z) are bounded on all ofR ⁿ (for e-nets in the case of sufficiently small ). It is shown that if E is close in a certain sense either to a relatively dense subset ofR ⁿ , or to an -net, then f(z) cannot increase fast alongR ⁿ . Similar estimates are established for integral metrics.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 74–76, 1988.