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The subject might be considered to be founded by the result of 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'). After that generalisation was made to simultaneous approximation, by Schmidt. 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 appproximation 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 theoryIn mathematics, a transcendental number is any irrational number that is not an algebraic number, i. it is not the solution of any polynomial equation of the form : where n ≥ 1 and the coefficients a are integers (or, equivalently, rationals), not all. Related to uniform distribution is the topic of irregularities of distribution , which is of a combinatorialCombinatorics Discrete mathematics Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with "counting" the objects in those collections enumerative combinatori nature. There are still simply-stated unsolved problems remaining in Diophantine approximation, for example Littlewood's conjecture.

See also: low-discrepancy sequenceIn mathematics, a low-discrepancy sequence is a sequence with the property that for all N the subsequence x . x is almost uniformly distributed (in a sense to be made precise), and x . x is almost uniformly distributed as well. The reader may wish to cons.