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A NOTE ON AN ASYMPTOTIC VERSION OF A PROBLEM OF MAHLER

Part of: Diophantine approximation, transcendental number theory Series expansions

Published online by Cambridge University Press:  15 September 2022

RICARDO FRANCISCO Open the ORCID record for RICARDO FRANCISCO [Opens in a new window]  and
DIEGO MARQUES Open the ORCID record for DIEGO MARQUES [Opens in a new window]
RICARDO FRANCISCO
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brasília, DF, Brazil e-mail: r.f.d.silva@mat.unb.br
DIEGO MARQUES*
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brasília, DF, Brazil
*
e-mail: diego@mat.unb.br
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Abstract

We prove that for any infinite sets of nonnegative integers $\mathcal {A}$ and $\mathcal {B}$ , there exist transcendental analytic functions $f\in \mathbb {Z}\{z\}$ whose coefficients vanish for any indexes $n\not \in \mathcal {A}+\mathcal {B}$ and for which $f(z)$ is algebraic whenever z is algebraic and $|z|<1$ . As a consequence, we provide an affirmative answer for an asymptotic version of Mahler’s problem A.

Keywords

Mahler problemtranscendental functionnatural asymptotic density

MSC classification

Primary: 11J81: Transcendence (general theory)
Secondary: 30B10: Power series (including lacunary series)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 3 , June 2023 , pp. 398 - 402
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The authors are supported by National Council for Scientific and Technological Development, CNPq.

References

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