If is an irrational algebraic number and is arbitrarily small, then there are only finitely many integer solutions and ( and being co-prime) of the inequality

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 is an algebraic number of degree , then the inequality has only finitely many integer solutions and ( and being co-prime) when . Siegel established that Thue's theorem is true for . The final version of the theorem stated above was obtained by Roth. There is a -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 -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)

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 -adic valuations as well. The latter work has profound consequences in the theory of exponential Diophantine equations (-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 -adic Thue–Siegel–Roth–Schmidt theorem"  Arch. Math. , 29  (1977)  pp. 267–270
[a3]	J.H. Evertse,   "On sums of -units and linear recurrences"  Compos. Math. , 53  (1984)  pp. 225–244
[a4]	G. Faltings,   "Diophantine approximation on abelian varieties"  Ann. of Math.  (Forthcoming)