Thue-Siegel-Roth theorem

If $\alpha$ is an irrational algebraic number and $\delta>0$ is arbitrarily small, then there are only finitely many integer solutions $p$ and $q>0$ ($p$ and $q$ being co-prime) of the inequality

$$\left|\alpha-\frac pq\right|<\frac{1}{q^{2+\delta}}.$$

This theorem is best possible of its kind; the number 2 in the exponent cannot be decreased. The Thue–Siegel–Roth theorem is a strengthening of the Liouville theorem (see Liouville number). Liouville's result has been successively strengthened by A. Thue [1], C.L. Siegel [2] and, finally, K.F. Roth [3]. Thue proved that if $\alpha$ is an algebraic number of degree $n\geq3$, then the inequality

$$\left|\alpha-\frac pq\right|<\frac{1}{q^\nu}$$

has only finitely many integer solutions $p$ and $q>0$ ($p$ and $q$ being co-prime) when $\nu>(n/2)+1$. Siegel established that Thue's theorem is true for $\nu>2n^{1/2}$. The final version of the theorem stated above was obtained by Roth. There is a $p$-adic analogue of the Thue–Siegel–Roth theorem. The results listed above are proved by non-effective methods (see Diophantine approximation, problems of effective).

References

[1]	A. Thue, "Bemerkungen über gewisse Annäherungsbrüche algebraischer Zahlen" Norske Vidensk. Selsk. Skrifter. , 3 (1908) pp. 1–34
[2]	C.L. Siegel, "Approximation algebraischer Zahlen" Math. Z. , 10 (1921) pp. 173–213
[3]	K.F. Roth, "Rational approximation to algebraic numbers" Mathematika , 2 : 1 (1955) pp. 1–20
[4]	K. Mahler, "Lectures on Diophantine approximations" , 1 , Univ. Notre Dame (1961)
[5]	D. Ridout, "The $p$-adic generalization of the Thue–Siegel–Roth theorem" Mathematika , 5 (1958) pp. 40–48
[6]	A.O. Gel'fond, "Transcendental and algebraic numbers" , Dover, reprint (1960) (Translated from Russian)

Comments

In 1971, W.M. Schmidt [a1] generalized Roth's theorem to the problem of simultaneous approximation of several algebraic numbers. This was extended by H.P. Schlickewei [a2] to include $p$-adic valuations as well. The latter work has profound consequences in the theory of exponential Diophantine equations ($S$-unit equations), see [a3].

In a completely different but spectacular direction, G. Faltings [a4] extended Roth's theorem to products of Abelian varieties and proved a remarkable conjecture of S. Lang on rational points on subvarieties of Abelian varieties. This proof can also be considered as a generalization of Vojta's proof of the Mordell conjecture (see also Thue–Siegel–Roth theorem).

References

[a1]	W.M. Schmidt, "Diophantine Approximation" , Lect. notes in math. , 785 , Springer (1980)
[a2]	H.P. Schlickewei, "The $p$-adic Thue–Siegel–Roth–Schmidt theorem" Arch. Math. , 29 (1977) pp. 267–270
[a3]	J.H. Evertse, "On sums of $S$-units and linear recurrences" Compos. Math. , 53 (1984) pp. 225–244
[a4]	G. Faltings, "Diophantine approximation on abelian varieties" Ann. of Math. (Forthcoming)