Some results on asymptotic versions of Mahler's problems

TL;DR

This paper constructs transcendental functions with bounded coefficients that take algebraic values at algebraic points, advancing the understanding of Mahler's problems and exceptional sets in transcendence theory.

Contribution

It demonstrates the existence of such functions and characterizes certain algebraic number subsets as exceptional sets, providing new insights into asymptotic Mahler-type problems.

Findings

Existence of transcendental functions with bounded coefficients taking algebraic values at algebraic points

Identification of exceptional subsets of algebraic numbers for these functions

Advancement in asymptotic versions of Mahler's problems

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.