An Ultrametric lethargy result and its application to p-adic number theory

TL;DR

This paper establishes a lethargy result in ultrametric approximation theory and demonstrates the existence of p-adic transcendental numbers with slow approximation error decay by algebraic p-adic numbers.

Contribution

It introduces a new lethargy result in ultrametric spaces and applies it to prove the existence of specific p-adic transcendental numbers.

Findings

Existence of p-adic transcendental numbers with slow approximation decay

Extension of lethargy results to non-Archimedean ultrametric contexts

Application to algebraic approximation in p-adic number theory

Abstract

In this paper we show a lethargy result in the non-Arquimedian context, for general ultrametric approximation schemes and, as a consequence, we prove the existence of p-adic transcendental numbers whose best approximation errors by algebraic p-adic numbers of degree less than or equal to n decays slowly.