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Diophantine approximation

In number theory, the field of Diophantine approximation deals with quantitative matters concerning the approximation of real numbers by rational numbers. Approximations exist that are as accurate as we choose; to quantify we can use the size of denominator required as a measure.

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+epsilon'). 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 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.