# Simultaneous approximation to values of the exponential function over the adeles

## TL;DR

This paper demonstrates that Hermite's approximations to exponential function values at algebraic numbers are nearly optimal from an adelic viewpoint, considering all completions of a number field.

## Contribution

It introduces an adelic framework to evaluate the optimality of Hermite's approximations across different completions of number fields.

## Key findings

- Hermite's approximations are nearly optimal adelically.
- The approach accounts for both Archimedean and p-adic valuations.
- Provides a unified adelic perspective on exponential value approximations.

## Abstract

We show that Hermite's approximations to values of the exponential function at given algebraic numbers are nearly optimal when considered from an adelic perspective. We achieve this by taking into account the ratio of these values whenever they make sense in the various completions (Archimedean or $p$-adic) of a number field containing these algebraic numbers.