Some results on asymptotic versions of Mahler's problems

TL;DR

This work advances Mahler's problems on the arithmetic of transcendental functions by constructing uncountably many $f\in\mathbb{Z}\{z\}$ with coefficients almost all bounded such that $f^{(m)}(\alpha)\in\overline{\mathbb{Q}}$ for all $m\ge0$ and algebraic $\alpha$ in the unit disk, and by realizing prescribed exceptional sets via sumsets $\mathcal{S}=\mathcal{A}+\mathcal{B}$. The core method combines a key lemma that produces $Q_k$ to force coefficients into $\mathcal{S}$ with lacunary series, ensuring analyticity and algebraicity of derivatives at algebraic points; this yields uncountably many such functions. The authors extend these constructions to control exceptional sets $S_f$ and density statements $\delta(L_0(f))$, and show how composing with a polynomial $P$ transfers exceptional sets while preserving or improving density bounds, thus addressing asymptotic variants of Mahler's problems and related C-problems. They obtain partial results for Problem C and connect to Tao–Ziegler partial-sum phenomena to ensure the relevance of the $\mathcal{A}+\mathcal{B}$ framework. Overall, the paper provides a versatile toolkit for generating transcendental functions with finely tuned arithmetic values and exceptional sets, with broad implications for arithmetic dynamics of power series.

Abstract

In this paper, we show the existence of a transcendental function $f\in\mathbb{Z}\{z\}$ with coefficients that are almost all bounded such that $f$ and all its derivatives assume algebraic values at algebraic points. Furthermore, we demonstrate that certain subsets of algebraic numbers are exceptional sets of some transcendental function $f\in\mathbb{Z}\{z\}$ with almost all bounded coefficients.

Theorems & Definitions (11)

• Theorem 1.1
• Corollary 1.1
• Corollary 1.2
• Theorem 1.2
• Corollary 1.3
• Theorem 1.3
• Theorem 1.4
• Corollary 1.4
• Lemma 2.1
• proof