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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t0927801.png" /> is an irrational [[Algebraic number|algebraic number]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t0927802.png" /> is arbitrarily small, then there are only finitely many integer solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t0927803.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t0927804.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t0927805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t0927806.png" /> being co-prime) of the inequality
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{{TEX|done}}
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If $\alpha$ is an irrational [[Algebraic number|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t0927807.png" /></td> </tr></table>
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$$\left|\alpha-\frac pq\right|<\frac{1}{q^{2+\delta}}.$$
  
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 number]]). Liouville's result has been successively strengthened by A. Thue [[#References|[1]]], C.L. Siegel [[#References|[2]]] and, finally, K.F. Roth [[#References|[3]]]. Thue proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t0927808.png" /> is an algebraic number of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t0927809.png" />, then 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 number]]). Liouville's result has been successively strengthened by A. Thue [[#References|[1]]], C.L. Siegel [[#References|[2]]] and, finally, K.F. Roth [[#References|[3]]]. Thue proved that if $\alpha$ is an algebraic number of degree $n\geq3$, then the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t09278010.png" /></td> </tr></table>
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$$\left|\alpha-\frac pq\right|<\frac{1}{q^\nu}$$
  
has only finitely many integer solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t09278011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t09278012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t09278013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t09278014.png" /> being co-prime) when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t09278015.png" />. Siegel established that Thue's theorem is true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t09278016.png" />. The final version of the theorem stated above was obtained by Roth. There is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t09278017.png" />-adic analogue of the Thue–Siegel–Roth theorem. The results listed above are proved by non-effective methods (see [[Diophantine approximation, problems of effective|Diophantine approximation, problems of effective]]).
+
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|Diophantine approximation, problems of effective]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Thue,  "Bemerkungen über gewisse Annäherungsbrüche algebraischer Zahlen"  ''Norske Vidensk. Selsk. Skrifter.'' , '''3'''  (1908)  pp. 1–34</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.L. Siegel,  "Approximation algebraischer Zahlen"  ''Math. Z.'' , '''10'''  (1921)  pp. 173–213</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K.F. Roth,  "Rational approximation to algebraic numbers"  ''Mathematika'' , '''2''' :  1  (1955)  pp. 1–20</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K. Mahler,  "Lectures on Diophantine approximations" , '''1''' , Univ. Notre Dame  (1961)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D. Ridout,  "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t09278018.png" />-adic generalization of the Thue–Siegel–Roth theorem"  ''Mathematika'' , '''5'''  (1958)  pp. 40–48</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.O. Gel'fond,  "Transcendental and algebraic numbers" , Dover, reprint  (1960)  (Translated from Russian)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Thue,  "Bemerkungen über gewisse Annäherungsbrüche algebraischer Zahlen"  ''Norske Vidensk. Selsk. Skrifter.'' , '''3'''  (1908)  pp. 1–34</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.L. Siegel,  "Approximation algebraischer Zahlen"  ''Math. Z.'' , '''10'''  (1921)  pp. 173–213</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K.F. Roth,  "Rational approximation to algebraic numbers"  ''Mathematika'' , '''2''' :  1  (1955)  pp. 1–20</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K. Mahler,  "Lectures on Diophantine approximations" , '''1''' , Univ. Notre Dame  (1961)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D. Ridout,  "The $p$-adic generalization of the Thue–Siegel–Roth theorem"  ''Mathematika'' , '''5'''  (1958)  pp. 40–48</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.O. Gel'fond,  "Transcendental and algebraic numbers" , Dover, reprint  (1960)  (Translated from Russian)</TD></TR></table>
  
  
  
 
====Comments====
 
====Comments====
In 1971, W.M. Schmidt [[#References|[a1]]] generalized Roth's theorem to the problem of simultaneous approximation of several algebraic numbers. This was extended by H.P. Schlickewei [[#References|[a2]]] to include <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t09278019.png" />-adic valuations as well. The latter work has profound consequences in the theory of exponential Diophantine equations (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t09278020.png" />-unit equations), see [[#References|[a3]]].
+
In 1971, W.M. Schmidt [[#References|[a1]]] generalized Roth's theorem to the problem of simultaneous approximation of several algebraic numbers. This was extended by H.P. Schlickewei [[#References|[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 [[#References|[a3]]].
  
 
In a completely different but spectacular direction, G. Faltings [[#References|[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|Mordell conjecture]] (see also Thue–Siegel–Roth theorem).
 
In a completely different but spectacular direction, G. Faltings [[#References|[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|Mordell conjecture]] (see also Thue–Siegel–Roth theorem).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.M. Schmidt,  "Diophantine Approximation" , ''Lect. notes in math.'' , '''785''' , Springer  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.P. Schlickewei,  "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t09278021.png" />-adic Thue–Siegel–Roth–Schmidt theorem"  ''Arch. Math.'' , '''29'''  (1977)  pp. 267–270</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.H. Evertse,  "On sums of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092780/t09278022.png" />-units and linear recurrences"  ''Compos. Math.'' , '''53'''  (1984)  pp. 225–244</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Faltings,  "Diophantine approximation on abelian varieties"  ''Ann. of Math.''  (Forthcoming)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.M. Schmidt,  "Diophantine Approximation" , ''Lect. notes in math.'' , '''785''' , Springer  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.P. Schlickewei,  "The $p$-adic Thue–Siegel–Roth–Schmidt theorem"  ''Arch. Math.'' , '''29'''  (1977)  pp. 267–270</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.H. Evertse,  "On sums of $S$-units and linear recurrences"  ''Compos. Math.'' , '''53'''  (1984)  pp. 225–244</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Faltings,  "Diophantine approximation on abelian varieties"  ''Ann. of Math.''  (Forthcoming)</TD></TR></table>

Revision as of 15:07, 2 August 2014

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)
How to Cite This Entry:
Thue-Siegel-Roth theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thue-Siegel-Roth_theorem&oldid=23085
This article was adapted from an original article by S.V. Kotov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article