论文标题
$ b'$
$B'$
论文作者
论文摘要
令$ n \ ge 2 $为整数，$α_1，\ ldots，α_n$为非零代数数。令$ b_1，\ ldots，b_n $是$ b_n \ not = 0 $的整数，并设置$ b = \ max \ {3，| b_1 |，\ ldots，| b_n | \} $。对于$ j = 1，\ ldots，n $，set $ h^*（α_j）= \ max \ {h（α_j），1 \} $，其中$ h $表示（boolegarithmic）weil高度。假设数量$λ= b_1 \logα_1 + \ cdots + b_n \logα_n$是非零的。 $ \ log |λ| $的典型下限，由贝克（Baker）在对数中的线性形式理论给出的形状$$ \ log |λ| \ ge -c（n，d）\，h^*（α_1）\ cdots h^*（α_n）\ log b，$ $，其中$ c（n，d）$是正面的，有效地计算，仅取决于$ n $和$ n $和$ n $ $ $ d $ $ n $ d $，由$α_1，\ ldots，\ ldots，\ ldots，\ ldots，α__n$。但是，在某些特殊情况下，尤其是$ | b_n | = 1 $，该界限可以改进到$$ \ log |λ| -c（n，d）\，h^*（α_1）\ cdots h^*（α_n）\ log \ frac {b} {h^*（α_n）}。 $$ $ b / h^*（α_n）$代替$ b $来源于费尔德曼和贝克的作品，是以有效的方式改进的关键工具，是真正代数数量的非理性指数至少$ 3 $的上限。我们对以代数数进行评估的近似指数的各种应用，对某些整数序列的$ s $部分以及对二磷酸方程式进行了调查。我们以关于收敛到实数的算术特性的一些新结果结束。
Let $n \ge 2$ be an integer and $α_1, \ldots, α_n$ be non-zero algebraic numbers. Let $b_1, \ldots , b_n$ be integers with $b_n \not= 0$, and set $B = \max\{3, |b_1|, \ldots , |b_n|\}$. For $j =1, \ldots, n$, set $h^* (α_j) = \max\{h(α_j), 1\}$, where $h$ denotes the (logarithmic) Weil height. Assume that the quantity $Λ= b_1 \log α_1 + \cdots + b_n \log α_n$ is nonzero. A typical lower bound of $\log |Λ|$ given by Baker's theory of linear forms in logarithms takes the shape $$ \log |Λ| \ge - c(n, D) \, h^* (α_1) \cdots h^* (α_n) \log B, $$ where $c(n,D)$ is positive, effectively computable and depends only on $n$ and on the degree $D$ of the field generated by $α_1, \ldots , α_n$. However, in certain special cases and in particular when $|b_n| = 1$, this bound can be improved to $$ \log |Λ| - c(n, D) \, h^* (α_1) \cdots h^* (α_n) \log \frac{B}{h^* (α_n)}. $$ The term $B / h^* (α_n)$ 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.
