View source for Mahler problem

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A conjecture in the [[Diophantine approximation, metric theory of|metric theory of Diophantine approximation]] stated by K. Mahler [[#References|[1]]]: For almost all (in the sense of [[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 [[#References|[2]]]. He also proved similar results for complex and $p$-adic numbers, and also for power series over finite fields.

====References====
<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 {{ZBL|0003.24602}}</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) {{ZBL|0181.05502}}</TD></TR>
</table>



====Comments====
The original paper of Sprindzhuk is [[#References|[a1]]].

====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 $S$ numbers"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''29'''  (1965)  pp. 379–436 {{ZBL|0156.05405}}</TD></TR>
</table>