克劳斯·罗特Klaus Roth

克劳斯·罗特 年轻时来到英国，1939年至1943年在伦敦的圣保罗学校就读。随后他前往剑桥大学彼得豪斯学院，于1945年获得学士学位。

毕业后，罗特 被任命为国际知名的戈登斯敦学校的助理教师，该校位于苏格兰埃尔金以北10公里处。这所学校由德国教育家库尔特·哈恩于1934年创立，是一所男校，强调在学术卓越之外还要发展品格。男孩们被期望生活在相当艰苦的条件下，没有任何生活奢侈品。

罗特 于1946年返回伦敦，在大学学院进行研究。他于1948年获得硕士学位，并于同年被任命为那里的助理讲师。两年后他获得博士学位，成为讲师，然后在1956年成为高级讲师，之后在1961年成为教授。

事实上，罗特 在大学学院担任讲师期间就取得了非凡的数学突破。他在 1955 年解决了用 有理的 逼近 代数数 这一重大未决问题。正是由于这项工作，罗特 于 1958 年被授予 菲尔兹奖。

对于任何无理数数$r$，很容易看出存在无穷多个有理数$\Large\frac{a}{b}\normalsize$满足

$| \Large\frac{a}{b}\normalsize - r | < \Large\frac{1}{b^{2}\normalsize}$

（$r$的连分数的所有收敛分数都满足这一点）。对于给定的$r$，令$\mu(r)$为指数$e$的上界，使得存在无穷多个有理数$\Large\frac{a}{b}\normalsize$满足

$| \Large\frac{a}{b}\normalsize - r | < \Large\frac{1}{b^{e}\normalsize}$。

上述表明$\mu(r) ≥ 2$对于所有$r$。
约瑟夫·刘维尔在1844年证明了如果r是次数为$n$的代数数，则$\mu(r) ≤ n$。
当时的未解决问题是找出在范围
$2 ≤ \mu(r) ≤ n$中，次数为$n$的代数数的$\mu(r)$值是多少。
阿克塞尔·图厄在1908年证明了$\mu(r) ≤ \Large\frac{n}{2}\normalsize + 1$，卡尔·西格尔在1921年将其改进为$\mu(r) ≤ 2√n$。
罗特在1955年通过证明对于任何代数数$r, \mu(r) = 2$完全解决了这个问题。

哈罗德·达文波特在1958年爱丁堡国际数学家大会上向罗特颁发了菲尔兹奖。谈到罗特对逼近代数数这一问题的解决时，哈罗德·达文波特说[2]：-

这一成就本身就能说明一切：它结束了一个篇章，现在又开启了一个新篇章。罗特的定理解决了一个既具有根本性质又极其困难的问题。只要数学还在被耕耘，它就将作为数学中的一座里程碑而存在。

哈罗德·达文波特在颁发约翰·查尔斯·菲尔兹奖章时，提到了罗特解决的另一个问题。这就是罗特在1952年对埃尔德什和图兰·帕尔于1935年提出的一个猜想的证明。该猜想涉及一个序列

$n_{1}, n_{2}, n_{3}, ...$

自然数满足

$n_{p} + n_{q} ≠ 2n_{r}$

除非$p = q = r$。如果$N(x)$表示该序列中小于$x$的项数，那么罗特证明了当$x \rightarrow ∞$时$\Large\frac{N(x)}{x}\normalsize \rightarrow 0$这一猜想。

哈罗德·达文波特在结束他的演讲[2]时说：-

《爱丽丝梦游仙境》中的公爵夫人说过，只要你能找到，凡事都有寓意。在罗特博士的工作中找到寓意并不困难。那就是，数学中那些重大的未解决问题，无论看起来多么困难、多么令人望而生畏，无论人们已经投入了多少努力，仍然可能因直接进攻而被攻克。

罗特于1966年转到伦敦帝国理工学院纯数学讲席，并一直担任该讲席至1988年。那一年，他前往伦敦大学学院担任访问教授，一直担任该职位到1996年，之后他回到苏格兰北部，离他刚开始研究生涯前在戈登斯敦学校任教的地方不远。

菲尔兹 奖章并不是授予 罗特 的唯一荣誉。他还获得许多其他荣誉，包括 1960 年当选 伦敦皇家学会 会士和 1993 年当选爱丁堡皇家学会会士。他于 1983 年获得 伦敦数学会 的 奥古斯塔斯·德摩根 奖章，并于 1991 年获得 皇家学会 的 詹姆斯·约瑟夫·西尔维斯特 奖章。

Klaus Roth came to England when he was young and attended St Paul's School in London from 1939 to 1943. He then went to Peterhouse, Cambridge where he was awarded his BA in 1945.

After graduating, Roth was appointed as an assistant master at the internationally famous Gordonstoun School, which lies 10 km north of Elgin in Scotland. The school had been founded in 1934 by the German educator Kurt Hahn as a boys' school that would emphasise the development of character in addition to academic excellence. The boys were expected to live in quite hard conditions without any of the luxuries of life.

Roth returned to London in 1946 to undertake research at University College. He was awarded his master's degree in 1948 and appointed an assistant lecturer there in that year. He was awarded his doctorate two years later, becoming a lecturer, then a reader in 1956, and then a professor in 1961.

In fact Roth made a remarkable mathematical breakthrough while still a lecturer at University College. He solved the major open problem of approximating algebraic numbers by rationals in 1955. It was for this work that Roth was awarded a Fields Medal in 1958.

For any irrational number $r$ it is easy to see that there are infinitely many rational numbers $\Large\frac{a}{b}\normalsize$ with

$| \Large\frac{a}{b}\normalsize - r | < \Large\frac{1}{b^{2}\normalsize}$

(the convergents for the continued fraction of $r$ all satisfy this). For a given $r$ let $\mu(r)$ be the upper bound the exponents $e$ such that there are infinitely many rational numbers $\Large\frac{a}{b}\normalsize$ with

$| \Large\frac{a}{b}\normalsize - r | < \Large\frac{1}{b^{e}\normalsize}$.

The above shows that $\mu(r) ≥ 2$for all $r$.
Liouville showed in 1844 that if r is an algebraic number of degree $n$ then $\mu(r) ≤ n$.
The open question was then to find where in the range
$2 ≤ \mu(r) ≤ n$ the value of $\mu(r)$ was for an algebraic number of degree $n$.
Thue showed that $\mu(r) ≤ \Large\frac{n}{2}\normalsize + 1$ in 1908 and Siegel improved this in 1921 to $\mu(r) ≤ 2√n$.
Roth solved the problem completely in 1955 by showing that for any algebraic number $r, \mu(r) = 2$.

Davenport presented Roth with the Fields Medal at the International Congress in Edinburgh in 1958. Speaking of Roth's solution to this problem of approximating algebraic numbers Davenport said [2]:-

The achievement is one that speaks for itself: it closes a chapter, and a new chapter is now opened. Roth's theorem settles a question which is both of a fundamental nature and of extreme difficulty. It will stand as a landmark in mathematics for as long as mathematics is cultivated.

Davenport, in his Fields Medal presentation, mentions another problem solved by Roth. This was Roth's proof in 1952 of a conjecture made in 1935 by Erdős and Turán. The conjecture concerned a sequence

$n_{1}, n_{2}, n_{3}, ...$

of natural numbers satisfying

$n_{p} + n_{q} ≠ 2n_{r}$

unless $p = q = r$. If $N(x)$ denotes the number of terms of the sequence less than $x$, Roth proved the conjecture that $\Large\frac{N(x)}{x}\normalsize \rightarrow 0$ as $x \rightarrow ∞$.

Davenport ends his address [2] by saying:-

The Duchess, in Alice in Wonderland, said that there is a moral in everything if only you can find it. It is not difficult to find the moral in Dr Roth's work. It is that the great unsolved problems of mathematics may still yield to direct attack however difficult and forbidding they appear to be, and however much effort has already been spent on them.

Roth moved to the chair of Pure Mathematics in Imperial College, London in 1966 and held this chair until 1988. In that year he went to University College, London as a visiting professor, a position he held until 1996 when he returned to the north of Scotland, not far from where he taught at Gordonstoun School before he began his research career.

The Fields Medal was not the only honour to be bestowed on Roth. He received many other honours including fellowship of the Royal Society of London in 1960 and of the Royal Society of Edinburgh in 1993. He received the De Morgan Medal of the London Mathematical Society in 1983 and the Sylvester Medal of the Royal Society in 1991.