Abstract
We examine the boundary behaviour of the generic power series f with coefficients chosen from a fixed bounded set Λ in the sense of Baire category. Notably, we prove that for any open subset U of the unit disk D with a nonreal boundary point on the unit circle, f(U) is a dense set of ℂ. As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property is given.
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Bolyai János Mathematical Society: Miklós Schweitzer Memorial Competition 2020, Problems and Solutions. Available at https://www.bolyai.hu/files/Schweitzer_2020_megoldasok.pdf. (In Hungarian.)
J. Breuer, B. Simon: Natural boundaries and spectral theory. Adv. Math. 226 (2011), 4902–4920.
J.-P. Kahane: Some Random Series of Functions. Cambridge Studies in Advanced Mathematics 5. Cambridge University Press, Cambridge, 1985.
J.-P. Kahane: Baire’s category theorem and trigonometric series. J. Anal. Math. 80 (2000), 143–182.
S. Kierst, E. Szpilrajn: Sur certaines singularités desfonctions analytiques uniformes. Fundam. Math. 21 (1933), 276–294. (In French.)
K. Kuratowski: Topology. Vol. 1. Academic Press, New York, 1966.
B. Maga, P. Maga: Random power series near the endpoint of the convergence interval. Publ. Math. 93 (2018), 413–424.
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We are grateful to the anonymous referees for their careful reading of the manuscript and their suggestions to improve the exposition.
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The first author was supported by the ÚNKP-20-3 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund, and by the Hungarian National Research, Development and Innovation, Grants 124003 and 124749. The second author was supported by the Premium Postdoctoral Fellowship of the Hungarian Academy of Sciences, the Rényi Intézet Lendület Automorphic Research Group and by the Hungarian National Research, Development and Innovation, Grants FK 135218, K 119528. Open access funding provided by Eötvös Loránd University.
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Maga, B., Maga, P. Generic power series on subsets of the unit disk. Czech Math J 72, 637–652 (2022). https://doi.org/10.21136/CMJ.2022.0021-21
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DOI: https://doi.org/10.21136/CMJ.2022.0021-21