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 *a _{1}*,

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*.