话语与超越
Words and Transcendence
摘要：
通过考虑在某些基$b\ge2$中的$b$元展开式，是否可能区分代数实数和超越实数？1950年，任。博雷尔建议，答案是否定的，并为任何真正的无理代数数$x$和任何基地$g\ge2$，$g$ -ary扩展$x$应满足一些法律，是共享的几乎所有数字。没有明确已知的三重$(g,a,x)$的例子，其中$g\ge3$是整数，$a$是$\{0,...,g-1\}$中的数字，而$x$是实无理代数数，因此人们可以声称数字$a$在$x$的$g$进制展开中无限频繁地出现。然而，最近取得了一些进展，主要归功于施密特子空间定理的巧妙使用。我们回顾其中的一些结果。
英文摘要：
Is it possible to distinguish algebraic from transcendental real numbers by considering the $b$-ary expansion in some base $b\ge2$? In 1950, É. Borel suggested that the answer is no and that for any real irrational algebraic number $x$ and for any base $g\ge2$, the $g$-ary expansion of $x$ should satisfy some of the laws that are shared by almost all numbers. There is no explicitly known example of a triple $(g,a,x)$, where $g\ge3$ is an integer, $a$ a digit in $\{0,...,g-1\}$ and $x$ a real irrational algebraic number, for which one can claim that the digit $a$ occurs infinitely often in the $g$-ary expansion of $x$. However, some progress has been made recently, thanks mainly to clever use of Schmidt's subspace theorem. We review some of these results.
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