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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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Correspondence to Adi Glücksam.

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Supported in part by ERC Advanced Grant 692616 and ISF Grant 382/15.

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Glücksam, A. Measurably entire functions and their growth. Isr. J. Math. 229, 307–339 (2019).

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