Proof of the transcendence of the Champernowne Constant with Thue-Siegel-Roth

Question

It is well know that the Champernowne Constant

0,1234567891011121314151617....

is transcendental. This was shown by Kurt Mahler in 1937. But the proof of the transcendence should also work with the famous Thue-Siegel-Roth theorem (http://en.wikipedia.org/wiki/Thue%E2%80%93Siegel%E2%80%93Roth_theorem), that was proved in 1955.

I'm looking for a reference of the transcendence-proof where this theorem is used.

Answer 1

Here is an extract from van der Poorten's "Obituary. Kurt Mahler (1903--1988)" (see p.353 in J. Austral. Math. Soc. Ser. A 51 (1991)):

In a more unexpected way, Mahler's arguments led to the following amusing result: Suppose $f$ is a non-constant polynomial taking integer values at the nonnegative integers. Then the concatenated decimal $$\phi=0.f(1)f(2)f(3)\dots $$ is transcendental. In particular Champernowne's normal number $$0.123\dots910111213\dots $$ is transcendental. Mahler's argument relies on the observation that one readily obtains rational approximations to $\phi$ with denominators high powers of the base 10, thus composed of the primes 2 and 5 alone. Perhaps disappointingly, Roth's definitive form of the Thue--Siegel inequalities permits a more immediate argument obviating the need for an appeal to the $p$-adic results.

This is to say that Roth's argument is more superior than Mahler's but it appeared some 20 years later...

Answer 2

There are applications of Thue--Siegel--Roth and related results to this question (but it seems that they use in some form or another the rational approximations already present in Mahler's work).

For example, Baker in "On Mahler's classification of transcendental numbers." Acta Math. 111 1964 97–120, shows that this constant is not a U-number using a result he describes as "This extends a theorem of LeVeque [..] which itself is a generalisation of Roth's Theorem"

However, then for the proof of this application it reads "It is clear from the proof of these results that the hypotheses of Theorem 1 are satisfied..." Where 'the proof of these results' refer to Mahler's.

And much more recently, Adamczewski and Bugeaud in "Mesures de transcendance et aspects quantitatifs de la méthode de Thue-Siegel-Roth-Schmidt." Proc. Lond. Math. Soc. 101 (2010), no. 1, 1–26 generalise this result of Baker together with a classical result of Ridout to then show that the generalizations of the Champernonwne Constant (any base, any suitable polynomial) are all S or T numbers. See section 3 of the paper, in particular Théorème 3.2. But again the proof uses the rational approximations of the original transcendence proofs (by Mahler); cf. the final paragraph of Section 3.

Answer 3

Let $M$ be Champernowne's number. Burger and Tubbs, Making Transcendence Transparent, Chapter 1, Sections 6 and 7, proves that there are infinitely many rational numbers $p_n/q_n$ satisfying $$\left|M-{p_n\over q_n}\right|<q_n^{-4.5}$$ Then they apply Roth's Theorem to prove $M$ is transcendental.