Approximation measures for shifted logarithms of algebraic numbers via effective Poincare-Perron

TL;DR

This paper develops new effective approximation measures for shifted logarithms of algebraic numbers in arbitrary number fields, refining previous measures and utilizing Poincare-Perron theorem for asymptotic estimates.

Contribution

It introduces novel approximation measures for shifted logarithms of algebraic numbers using Pade approximants and Poincare-Perron theorem, applicable to number fields of any degree.

Findings

Refined approximation measures for shifted logarithms.

Application of Poincare-Perron theorem for asymptotic estimates.

Extension of methods to general number fields.

Abstract

In this article, we show new effective approximation measures for the shifted logarithm of algebraic numbers which belong to a number field of arbitrary degree. Our measures refine previous ones including those for usual logarithms. We adapt Pade approximants constructed by M. Kawashima and A. Poels in the rational case. Our key ingredient relies on the Poincare-Perron theorem which gives us asymptotic estimates at every archimedean place of the number field, that we apply to decide whether the shifted logarithm lies outside of the number field. We use Perron's second theorem and its modification due to M. Pituk to handle general cases.