- Diophantine approximation
The absolute value of the difference between the real number to be approximated and the rational number that approximates it is a crude measure of how good the approximation is. However, since the rational numbers are dense in the real numbers, one can always find rational numbers that are arbitrarily close to the number being approximated. So this measure tells us nothing about the "quality" of the approximation.
A better measure of the quality of the approximation is by comparison of the difference to the size of the denominator.
E.g., 22/7 and 179/57 are roughly equally far from π, but number theorists would consider 22/7 to be a better approximation because its denominator is smaller.
Approximation to algebraic numbers
In the 1840s, Joseph Liouville obtained an important result on general algebraic numbers (the Lemma on the page for Liouville number). If x is an irrational algebraic number of degree n over the rational numbers, then there exists a constant c(x) > 0 such that
holds for all integers p and q where q > 0.
This result allowed him to produce the first proven examples of transcendental numbers. This link between diophantine approximation and transcendence theory continues to the present-day. Many of the proof techniques are shared between the two areas.
Liouville's 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 (e.g. 2 + ε). Subsequently, Wolfgang M. Schmidt generalised this to the case of simultaneous approximation. The proofs were difficult, and not effective. This means that we cannot use the results or their proofs to obtain bounds on the size of solutions of associated diophantine equations. However, the techniques and results can often be used to bound the number of solutions of such equations.
Note that Khinchine proved that if
is a non-increasing function and , then for almost all real numbers x (not necessarily algebraic), there are at most finitely many rational p/q with q not zero and
Similarly, if the sum diverges, then for almost all real numbers, there are infinitely many such rational numbers p/q.
In 1941, R.J. Duffin and A.C. Schaeffer  proved a more general theorem that implies Khinchine's result, and made a conjecture now known by their name as the Duffin–Schaeffer conjecture. In 2006, V. Beresnevich and S. Velani proved a Hausdorff measure analogue of the conjecture, published in the Annals of Mathematics.
There are several effective techniques and results available. The most general is lower bounds for linear forms in logarithms, which were developed by Alan Baker. A refinement of Baker's theorem by Fel'dman implied that if x is an algebraic number of degree n over the rational numbers, then there exist effectively computable constants c(x) > 0 and 0 < d(x) < n such that
holds for all rational integers with q not zero.
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 throughout analytic number theory in the bounding of error terms.
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 the Littlewood conjecture.
In his plenary address at the International Mathematical Congress in Kyoto (1990), Grigory Margulis outlined a broad program rooted in ergodic theory that allows one 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–Mozes, and the proof of Baker and Sprindzhuk conjectures in the Diophantine approximations on manifolds by Kleinbock and Margulis. Various generalizations of the above results of Aleksandr Khinchin in metric Diophantine approximation have also been obtained within this framework.
- ^ R. J. Duffin and A. C. Schaeffer, Khintchine's problem in metric Diophantine approximation, Duke Mathematical Journal, 8 (1941), 243–255
- ^ V. Beresnevich and S. Velani, A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures, Annals of Mathematics, 164 (2006), 971–992
- J.W.S. Cassels (1957). An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics. 45. Cambridge University Press.
- Kleinbock, D; Margulis, G (1998). "Flows on homogeneous spaces and Diophantine approximation on manifolds". Ann. Math. 148 (1): 339–360. doi:10.2307/120997. JSTOR 120997. MR1652916.
- Lang, S (1995). Introduction to Diophantine Approximations (New Expanded ed.). 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 ISBN 0-521-80799-9.
- Wolfgang M. Schmidt. Diophantine approximation. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])
- Wolfgang M. Schmidt.Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, Springer Verlag 2000
- Sprindzhuk, V (1979). Metric theory of Diophantine approximations. John Wiley & Sons, New York. ISBN 0-470-26706-2. MR0548467.
- Diophantine Approximation: historical survey. From Introduction to Diophantine methods course by Michel Waldschmidt.
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