# Thue-Siegel-Roth theorem

2010 Mathematics Subject Classification: Primary: 11J68 [MSN][ZBL]

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)

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].
 [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)