A question proposed by K. Mahler on exceptional sets of transcendental functions with integer coefficients: solution of a Mahler's problem

TL;DR

This paper proves that any subset of algebraic numbers in the unit disk, closed under conjugation and containing zero, can be realized as the exceptional set of uncountably many transcendental functions with integer coefficients, solving a longstanding problem.

Contribution

It provides a construction showing that a broad class of subsets can be realized as exceptional sets of transcendental functions, addressing Mahler's 1976 question.

Findings

Any such subset is the exceptional set of uncountably many transcendental functions.

These functions are analytic in the unit ball and have integer coefficients.

The result confirms the richness of transcendental functions with prescribed exceptional sets.

Abstract

In this paper, we shall prove that any subset of $\overline{\mathbb Q}\cap B(0,1)$, which is closed under complex conjugation and which contains the element $0$, is the exceptional set of uncountably many transcendental functions, analytic in the unit ball, with integer coefficients. This solves a strong version of an old question proposed by K. Mahler (1976).