Mahler problem

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