$B'$

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.