**Wolfgang Schmidt (1933- )**

**Schmidt's Theorem 1 (1967). Let **
** and **
** be real algebraic numbers such that **
** are linearly independent over the rational numbers, and let **
** > 0. Then there are only a finite number of (simultaneous) rational approximations **
** and **
** such that**

** and **

**Schmidt's Theorem 2 (1970). Let **
** be real algebraic numbers such that **
** are linearly independent over the rational numbers Q, and let **
** > 0. Then there are only a finite number of simultaneous rational approximations **
** such that**

**, ... , **

**I regret I have omitted Mahler's famous A-, S-, T-, U- classification of real and complex numbers, which I briefly mentioned in the Mahler section earlier. Suffice it to remark that algebraic numbers are of type A, while transcendental numbers fall into the other three categories: roughly those that **
**cannot be well-approximated**
** by algebraic numbers (those are the S-numbers), those that **
**can be well-approximated**
** by algebraic numbers (those are the U-numbers, which in turn are further classified according to their integral **
**degree**
**, and here the Liouville numbers are an extreme example: those of degree 1), and finally those as it were in between (the T-numbers).**

** Mahler himself proved that almost all (in the exact sense of Lebesgue measure theory) real or complex numbers are of S-type. There was then a long-standing conjecture of Mahler's - dating from 1932 - that almost all real numbers are S-numbers of **
**type **
**1, and almost all complex numbers are S-numbers of **
**type**
** **
**. That was settled in 1965 by Sprindzuk. In 1953 LeVeque proved the existence of U-numbers of each degree.**

** **

**Schmidt's fundamental contribution, in this connection, was to prove the **
**existence**
** of T-numbers. I quote from the Introduction to Schmidt's 1970 paper **
**T**
**-NUMBERS DO EXIST (from a 1968 conference): **
**K. Mahler in 1932 divided the real transcendental numbers into three classes, and called numbers in these classes S-numbers, T-numbers and U-numbers. But while the existence of S-numbers and of U-numbers is easy to see, the existence of T-numbers was left open. It is the purpose of the present paper to prove the following**

**Theorem 1. T-numbers do exist**
**.**

** **
**The proof will be via T*-numbers introduced by Koksma **
**[1939]. **
**We shall make essential use of a recent theorem of Wirsing about approximations to an algebraic number by algebraic numbers of a given degree.**