$B'$

TL;DR

This paper discusses improved lower bounds for linear forms in logarithms of algebraic numbers, with applications to Diophantine approximation, integer sequences, and properties of convergents, enhancing classical results with effective bounds.

Contribution

It introduces refined bounds for linear forms in logarithms in special cases, leading to better effective estimates in Diophantine approximation and related areas.

Findings

Improved lower bounds for linear forms in logarithms when certain coefficients are ±1.

Applications to effective bounds on irrationality exponents of algebraic numbers.

New results on the arithmetical properties of convergents to real numbers.

Abstract

Let $n \ge 2$ be an integer and $\alpha_1, \ldots, \alpha_n$ be non-zero algebraic numbers. Let $b_1, \ldots , b_n$ be integers with $b_n \not= 0$, and set $B = \max\{3, |b_1|, \ldots , |b_n|\}$. For $j =1, \ldots, n$, set $h^* (\alpha_j) = \max\{h(\alpha_j), 1\}$, where $h$ denotes the (logarithmic) Weil height. Assume that the quantity $\Lambda = b_1 \log \alpha_1 + \cdots + b_n \log \alpha_n$ is nonzero. A typical lower bound of $\log |\Lambda|$ given by Baker's theory of linear forms in logarithms takes the shape $$ \log |\Lambda| \ge - c(n, D) \, h^* (\alpha_1) \cdots h^* (\alpha_n) \log B, $$ where $c(n,D)$ is positive, effectively computable and depends only on $n$ and on the degree $D$ of the field generated by $\alpha_1, \ldots , \alpha_n$. However, in certain special cases and in particular when $|b_n| = 1$, this bound can be improved to $$ \log |\Lambda| - c(n, D) \, h^* (\alpha_1) \cdots h^* (\alpha_n) \log \frac{B}{h^* (\alpha_n)}. $$ The term $B / h^* (\alpha_n)$ in place of $B$ originates in works of Feldman and Baker and is a key tool for improving, in an effective way, the upper bound for the irrationality exponent of a real algebraic number of degree at least $3$ given by Liouville's theorem. We survey various applications of this refinement to exponents of approximation evaluated at algebraic numbers, to the $S$-part of some integer sequences, and to Diophantine equations. We conclude with some new results on arithmetical properties of convergents to real numbers.