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
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 . He also proved similar results for complex and $p$-adic numbers, and also for power series over finite fields.
|||K. Mahler, "Ueber das Mass der Menge aller $S$-Zahlen" Math. Ann. , 106 (1932) pp. 131–139 Zbl 0003.24602|
|||V.G. Sprindzhuk, "Mahler's problem in metric number theory" , Amer. Math. Soc. (1969) (Translated from Russian) Zbl 0181.05502|
The original paper of Sprindzhuk is [a1].
|[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 Zbl 0156.05405|
Mahler problem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Mahler_problem&oldid=38541