Illinois Journal of Mathematics

On the exceptional set in a conditional theorem of Littlewood

Lukas Geyer

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Abstract

In 1952, Littlewood stated a conjecture about the average growth of spherical derivatives of polynomials, and showed that it would imply that for entire function of finite order, “most” preimages of almost all points are concentrated in a small subset of the plane. In 1988, Lewis and Wu proved Littlewood’s conjecture. Using techniques from complex dynamics, we construct entire functions of finite order with a bounded set of singular values for which the set of exceptional preimages is infinite, with logarithmically growing cardinality.

Article information

Source
Illinois J. Math., Volume 58, Number 1 (2014), 279-284.

Dates
First available in Project Euclid: 1 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1427897178

Digital Object Identifier
doi:10.1215/ijm/1427897178

Mathematical Reviews number (MathSciNet)
MR3331851

Zentralblatt MATH identifier
1341.30028

Subjects
Primary: 30D35: Distribution of values, Nevanlinna theory
Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]

Citation

Geyer, Lukas. On the exceptional set in a conditional theorem of Littlewood. Illinois J. Math. 58 (2014), no. 1, 279--284. doi:10.1215/ijm/1427897178. https://projecteuclid.org/euclid.ijm/1427897178


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