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A conjecture in the metric theory of Diophantine approximation (cf. [[Diophantine approximation, metric theory of|Diophantine approximation, metric theory of]]) stated by K. Mahler [[#References|[1]]]: For almost-all (in the sense of the Lebesgue measure) numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062140/m0621401.png" /> the inequality
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A conjecture in the metric theory of Diophantine approximation (cf. [[Diophantine approximation, metric theory of|Diophantine approximation, metric theory of]]) stated by K. Mahler [[#References|[1]]]: For almost-all (in the sense of the Lebesgue measure) numbers $\omega\in\mathbf R$ 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/m/m062/m062140/m0621402.png" /></td> </tr></table>
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$$|P(\omega)|<|H(P)|^{-n-\epsilon}$$
  
has a finite number of solutions in polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062140/m0621403.png" /> of degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062140/m0621404.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062140/m0621405.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062140/m0621406.png" /> is a natural number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062140/m0621407.png" /> is the maximum modulus of the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062140/m0621408.png" />. An equivalent formulation is: For almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062140/m0621409.png" /> the inequality
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has a finite number of solutions in polynomials $P\in\mathbf Z[x]$ of degree not exceeding $n$. Here $\epsilon>0$, $n$ is a natural number and $H(P)$ is the maximum modulus of the coefficients of $P$. An equivalent formulation is: For almost-all $\omega\in\mathbf R$ 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/m/m062/m062140/m06214010.png" /></td> </tr></table>
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$$\max(\|\omega q\|,\ldots,\|\omega^nq\|)<q^{-1/n-\epsilon}$$
  
has a finite number of solutions in integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062140/m06214011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062140/m06214012.png" /> is the distance from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062140/m06214013.png" /> to the nearest integer).
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has a finite number of solutions in integers $q$ ($\|\alpha\|$ is the distance from $\alpha$ to the nearest integer).
  
Mahler's problem was solved affirmatively in 1964 by V.G. Sprindzhuk [[#References|[2]]]. He also proved similar results for complex and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062140/m06214014.png" />-adic numbers, and also for power series over finite fields.
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Mahler's problem was solved affirmatively in 1964 by V.G. Sprindzhuk [[#References|[2]]]. He also proved similar results for complex and $p$-adic numbers, and also for power series over finite fields.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Mahler,  "Ueber das Mass der Menge aller <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062140/m06214015.png" />-Zahlen"  ''Math. Ann.'' , '''106'''  (1932)  pp. 131–139</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.G. Sprindzhuk,  "Mahler's problem in metric number theory" , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Mahler,  "Ueber das Mass der Menge aller $S$-Zahlen"  ''Math. Ann.'' , '''106'''  (1932)  pp. 131–139</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.G. Sprindzhuk,  "Mahler's problem in metric number theory" , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.G. Sprindzhuk,  "A proof of Mahler's conjecture on the measure of the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062140/m06214016.png" /> numbers"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''29'''  (1965)  pp. 379–436</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.G. Sprindzhuk,  "A proof of Mahler's conjecture on the measure of the set of $S$ numbers"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''29'''  (1965)  pp. 379–436</TD></TR></table>

Revision as of 06:26, 15 August 2014

A conjecture in the metric theory of Diophantine approximation (cf. Diophantine approximation, metric theory of) stated by K. Mahler [1]: For almost-all (in the sense of the Lebesgue measure) numbers $\omega\in\mathbf R$ the inequality

$$|P(\omega)|<|H(P)|^{-n-\epsilon}$$

has a finite number of solutions in polynomials $P\in\mathbf Z[x]$ of degree not exceeding $n$. Here $\epsilon>0$, $n$ is a natural number and $H(P)$ is the maximum modulus of the coefficients of $P$. An equivalent formulation is: For almost-all $\omega\in\mathbf R$ the inequality

$$\max(\|\omega q\|,\ldots,\|\omega^nq\|)<q^{-1/n-\epsilon}$$

has a finite number of solutions in integers $q$ ($\|\alpha\|$ is the distance from $\alpha$ to the nearest integer).

Mahler's problem was solved affirmatively in 1964 by V.G. Sprindzhuk [2]. He also proved similar results for complex and $p$-adic numbers, and also for power series over finite fields.

References

[1] K. Mahler, "Ueber das Mass der Menge aller $S$-Zahlen" Math. Ann. , 106 (1932) pp. 131–139
[2] V.G. Sprindzhuk, "Mahler's problem in metric number theory" , Amer. Math. Soc. (1969) (Translated from Russian)


Comments

The original paper of Sprindzhuk is [a1].

References

[a1] V.G. Sprindzhuk, "A proof of Mahler's conjecture on the measure of the set of $S$ numbers" Izv. Akad. Nauk SSSR Ser. Mat. , 29 (1965) pp. 379–436
How to Cite This Entry:
Mahler problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mahler_problem&oldid=32949
This article was adapted from an original article by Yu.V. Nesterenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article