Measurably entire functions and their growth
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In 1997 B. Weiss introduced the notion of measurably entire functions and proved that they exist on every arbitrary free C-action defined on a standard probability space. In the same paper he asked about the minimal possible growth rate of such functions. In this work we show that for every arbitrary free C-action defined on a standard probability space there exists a measurably entire function whose growth rate does not exceed exp(exp[logp |z|]) for any p > 3. This complements a recent result by Buhovski, Glücksam, Logunov and Sodin who showed that such functions cannot have a growth rate smaller than exp(exp[logp |z|]) for any p < 2.
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- L. Buhovsky, A. Glücksam, A. Logunov and M. Sodin, Translation-invariant probability measures on entire functions, Journal d’Analyse Mathématique, to appear, arXiv:1703.08101.Google Scholar