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 the inequality
![]() |
has a finite number of solutions in polynomials of degree not exceeding
. Here
,
is a natural number and
is the maximum modulus of the coefficients of
. An equivalent formulation is: For almost-all
the inequality
![]() |
has a finite number of solutions in integers (
is the distance from
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 -adic numbers, and also for power series over finite fields.
References
[1] | K. Mahler, "Ueber das Mass der Menge aller ![]() |
[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 ![]() |
Mahler problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mahler_problem&oldid=32949