Kurt Mahler classified the complex numbers into fours categories. These are A-numbers, S-numbers, U-numbers and T-numbers. It is known that A-numbers are precisely the algebraic numbers. Also, Liouville's numbers are contained in the set of U-numbers. It is also known that all the set of numbers A, U, T are set of measure zero and hence S-numbers form a set of full measure. Does any one know example of S-number or T-number?
Mahler himself proved that
0.123456789101112131516...
is an S-number.
As far as I know, no explicit examples of T-numbers are known, even if it is known that they actually exist (thanks to Wolfgang Schmidt).
Dear Antonio M. Oller-Marcén
Thank you very much for the information. Can you provide me the paper where Mahler proved 0.123456789101112131516... is an S-number.
I am aware of Schmidt work but his proof is not constructual. In fact Schmidt proved the existence of Koksma's T* -numbers and it is easy to see T*-numbers are also T-numbers.
You can read the complete proof on one of my favourite books on trnascendental number theory, called “Making transcendence transparent: an intuitive approach to classical transcendental number theory“, by Edward Burger and Robert Tubbs, Section 1.6.
Dear Ana Paula Chaves ,
Thanks for your reply. Do you have a pdf copy of the book?
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