A Liouville-Type Inequality for Values of Mahler M-Functions

TL;DR

This paper proves a Liouville-type inequality for Mahler Mq-functions at algebraic points, showing these values are not Liouville or U-numbers, thus solving a longstanding problem.

Contribution

It establishes a new inequality for Mahler Mq-functions and resolves a major open problem regarding their values at algebraic points.

Findings

Values of Mahler Mq-functions are not Liouville numbers.

Values are not U-numbers.

The result applies to arbitrary Mahler Mq-functions at algebraic points.

Abstract

We establish a Liouville-type inequality for the values, at a common nonzero algebraic point, of arbitrary Mahler Mq-functions. As an application, we prove that no such value is a Liouville number, or even a U -number. This solves a long-standing problem in the field.