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On the exceptional set of transcendental functions with integer coefficients in a prescribed set: the problems A and C of Mahler. (English) Zbl 07257229
A transcendental function is a function \( f(x) \) such that the only complex polynomial \( P \) satisfying \( P(x, f(x)) =0 \), for all \( x \) in its domain, is the zero polynomial. Trigonometric functions, the exponential function, and their inverses are some of the examples of transcendental functions. Denote by \( \bar{\mathbb{Q}} \) the field of algebraic numbers. For a function \( f \) analytic in the complex domain \( \mathcal{D} \), define the exceptional set \( S_f \) of \( f \) as \(\displaystyle{S_f=\left\{ \alpha \in \bar{\mathbb{Q}}\cap \mathcal{D}: f(\alpha) \in \bar{\mathbb{Q}} \right\}}\). For example, the exceptional sets of the functions \( 2^{z} \) and \( e^{z\pi+1} \) are \( \mathbb{Q} \) and \( \emptyset \), respectively, as shown by the Gelfond-Schneider theorem and Baker’s theorem.
In the paper under review, the authors consider Problem A and Problem C in the book of K. Mahler [Lectures of transcendental numbers (1976; Zbl 0332.10019)], who suggested three problems, which he named Problem A, B and C, on the arithmetic behaviour of transcendental functions. Problems B and C have been completely solved by the authors in [Math. Ann. 368, No. 3–4, 1059–1062 (2017; Zbl 1387.11056); Bull. Aust. Math. Soc. 98, No. 1, 60–63 (2018; Zbl 1422.11162)], but Problem A remains open in general. Recall that, as usual, \( \mathbb{Z}{\{z\}} \) denotes the set of the power series analytic in the unit ball \( B(0,1) \) and with integer coefficients. Problems A and C are stated as follows.
Does there exist a transcendental function \( f\in \mathbb{Z}{\{z\}} \) with bounded coefficients and such that \( f(\bar{\mathbb{Q}}\cap B(0,1)) \subseteq \bar{\mathbb{Q}} \)?
Does there exist for every choice of \( S \) (closed under complex conjugation and such that \( 0\in S \)) a transcendental entire function with rational coefficients for which \( S_f=S \)?

In this paper, the authors generalize the main result of J. Haung, et al. [Bull. Aust. Math. Soc. 82, No. 2, 322–327 (2010; Zbl 1204.11113)]. As a consequence, the authors improve their main result in [Acta Arith. 192, No. 4, 313–327 (2020; Zbl 1450.11078)] as well as providing a variant version of Problem A (for coefficients belonging to some zero asymptotic density sets). Recall that an \( n \)-smooth integer is an integer (possibly negative) whose prime factors are all less than or equal to \( n \). The main result in this paper is the following.
Theorem. Let \( A \) be a countable subset of \( B(0,1) \) which is closed under complex conjugation. For each \( \alpha \in A \), fix a dense subset \( E_\alpha \subseteq \mathbb{C} \) (such that \( 0\in A \) , then \( 1\in E_0 \), \( E_\alpha \) is dense in \( \mathbb{R} \) whenever \( \alpha \in \mathbb{R} \), and such that \(\bar{E_\alpha} = E_{\bar{\alpha}}\), for all \( \alpha\in A \)). Then there exist uncountably many transcendental functions \( \displaystyle{f(z)=\sum_{n\ge 0}a_nz^{n} \in \mathbb{Z}\{z\}} \), such that \( a_n \) is a \( 3 \)-smooth number (for all \( n\ge 0 \)) and \( f(\alpha) \in E_\alpha \), for all \( \alpha \in A \).
11J81 Transcendence (general theory)
11J91 Transcendence theory of other special functions
30D20 Entire functions of one complex variable, general theory
11N05 Distribution of primes
30B10 Power series (including lacunary series) in one complex variable
Full Text: DOI
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