Title: Measurably entire functions and their growth,
Abstract: In 1997 B. Weiss introduced the notion of measurably entire functions and
proved that they exist on every arbitrary free C- action defined on standard
probability space. In the same paper he asked about the minimal possible growth
of measurably entire 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 does not exceed exp (exp[log^p |z|])
for any p > 3. This complements a recent result by Buhovski, Glücksam,
Logunov, and Sodin (arXiv:1703.08101) who showed that such functions cannot
grow slower than exp (exp[log^p |z|]) for any p < 2.