# Mahler problem

2020 Mathematics Subject Classification: Primary: 11J83 [MSN][ZBL]

A conjecture in the metric theory of Diophantine approximation stated by K. Mahler : 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 . He also proved similar results for complex and $p$-adic numbers, and also for power series over finite fields.

How to Cite This Entry:
Mahler problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mahler_problem&oldid=38541
This article was adapted from an original article by Yu.V. Nesterenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article