Diophantine approximation
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In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers. The smallness of the distance (in an absolute value sense) from the real number to be approximated to the rational number that approximates it is a crude measure of how good the approximation is. A subtler measure considers how good the approximation is by comparison to the size of the denominator.
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[edit] Classical approach
The subject can be seen as having been founded on a result of Joseph Liouville on general algebraic numbers (the Lemma on the page for Liouville number). Before that, much was known from the theory of continued fractions, as applied to square roots of integers and other quadratic irrationals.
This result was improved by Axel Thue and others, leading in the end to a definitive theorem of Roth: the exponent in the theorem was reduced from n, the degree of the algebraic number, to any number greater than 2 (i.e. '2+ε'). Subsequently, Schmidt generalised this to the case of simultaneous approximation. The proofs were difficult, and not effective, a disadvantage in applications.
Another topic that has seen a thorough development is the theory of uniform distribution mod 1. Take a sequence a1, a2, ... of real numbers and consider their fractional parts. That is, more abstractly, look at the sequence in R/Z, which is a circle. For any interval I on the circle we look at the proportion of the sequence's elements that lie in it, up to some integer N, and compare it to the proportion of the circumference occupied by I. Uniform distribution means that in the limit, as N grows, the proportion of hits on the interval tends to the 'expected' value. Hermann Weyl proved a basic result showing that this was equivalent to bounds for exponential sums formed from the sequence. This showed that Diophantine approximation results were closely related to the general problem of cancellation in exponential sums, which occurs all over analytic number theory in the bounding of error terms.
After Roth's theorem, the major advances in the subject have been in connection with transcendence theory. Related to uniform distribution is the topic of irregularities of distribution, which is of a combinatorial nature. There are still simply-stated unsolved problems remaining in Diophantine approximation, for example Littlewood's conjecture.
[edit] Recent developments
In his plenary address at the International Mathematical Congress in Kyoto (1990), Grigory Margulis outlined a broad program rooted in ergodic theory that allows to prove number-theoretic results using the dynamical and ergodic properties of actions of subgroups of semisimple Lie groups. The work of D.Kleinbock, G.Margulis, and their collaborators demonstrated the power of this novel approach to classical problems in Diophantine approximation. Among its notable successes are the proof of the decades-old Oppenheim conjecture by Margulis, with later extensions by Dani and Margulis and Eskin–Margulis–Moses, and the proof of Baker and Sprindzhuk conjectures in the Diophantine approximations on manifolds by Kleinbock and Margulis. Various generalizations of the results of Aleksandr Khinchin in metric Diophantine approximation have also been obtained within this framework.
[edit] References
- Kleinbock, D; Margulis, G (1998). "Flows on homogeneous spaces and Diophantine approximation on manifolds". Ann. of Math. 148 (1): 339–360. doi:. MR1652916.
- Lang, S (1995). Introduction to Diophantine Approximations, New Expanded Edition, Springer-Verlag. ISBN 0-387-94456-7.
- Grigory Margulis, Diophantine approximation, lattices and flows on homogeneous spaces. A panorama of number theory or the view from Baker's garden (Zürich, 1999), 280–310, Cambridge Univ. Press, Cambridge, 2002 MR1975458
- Sprindzhuk, V (1979). Metric theory of Diophantine approximations. John Wiley & Sons, New York. ISBN 0-470-26706-2 MR0548467.
