Effective irrationality measures for real and $p$-adic roots of rational numbers close to $1$, with an application to parametric families of Thue-Mahler equations

TL;DR

This paper develops effective irrationality measures for roots of rational numbers near 1 using linear forms in logarithms, extending to p-adic cases, and applies these results to solve specific Thue-Mahler equations.

Contribution

It introduces new effective irrationality bounds for roots of rationals close to 1, including p-adic analogues, and applies these to solve particular families of Thue-Mahler equations.

Findings

Effective irrationality measures for roots near 1.

p-adic analogue for roots close to p-adic integers.

Complete solutions for certain Thue-Mahler equations.

Abstract

We show how the theory of linear forms in two logarithms allows one to get effective irrationality measures for $n$-th roots of rational numbers ${a \over b}$, when $a$ is very close to $b$. We give a $p$-adic analogue of this result under the assumption that $a$ is $p$-adically very close to $b$, that is, that a large power of $p$ divides $a-b$. As an application, we solve completely certain families of Thue-Mahler equations. Our results illustrate, admittedly in a very special situation, the strength of the known estimates for linear forms in logarithms.