Values of E-functions are not Liouville numbers

TL;DR

This paper proves that the value of an $E$-function at an algebraic point can never be a Liouville number, extending previous measures and solving a longstanding problem in transcendence theory.

Contribution

It establishes an analogue of Shidlovskii's measure for $E$-functions at algebraic points, showing these values are not Liouville numbers and clarifying linear independence over algebraic and rational fields.

Findings

Values of $E$-functions at algebraic points are not Liouville numbers.

Linear independence over algebraic closure is equivalent to over rationals for $E$-function values.

Extended previous results using improvements in $E$-operator theory.

Abstract

Shidlovskii has given a linear independence measure of values of $E$-functions with rational Taylor coefficients at a rational point, not a singularity of the underlying differential system satisfied by these $E$-functions. Recently, Beukers has proved a qualitative linear independence theorem for the values at an algebraic point of $E$-functions with arbitrary algebraic Taylor coefficients. In this paper, we obtain an analogue of Shidlovskii's measure for values of arbitrary $E$-functions at algebraic points. This enables us to solve a long standing problem by proving that the value of an $E$-function at an algebraic point is never a Liouville number. We also prove that values at rational points of $E$-functions with rational Taylor coefficients are linearly independent over $\overline{\mathbb{Q}}$ if and only if they are linearly independent over $\mathbb{Q}$. Our methods rest upon improvements of results obtained by Andr\'e and Beukers in the theory of $E$-operators.