The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number a/b is a "good" approximation of a real number α if the absolute value of the difference between a/b and α may not decrease if a/b is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of continued fractions.
Knowing the "best" approximations of a given number, the main problem of the field is to find sharp upper and lower bounds of the above difference, expressed as a function of the denominator. It appears that these bounds depend on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number is larger than the lower bound for algebraic numbers, which is itself larger than the lower bound for all real numbers. Thus a real number that may be better approximated than the bound for algebraic numbers is certainly a transcendental number.
Diophantine approximations and transcendental number theory are very close areas that share many theorems and methods. Diophantine approximations also have important applications in the study of Diophantine equations.
Given a real number α, there are two ways to define a best Diophantine approximation of α. For the first definition, the rational number p/q is a best Diophantine approximation of α if
for every rational number p'/q' different from p/q such that 0 < q′ ≤ q.
A best approximation for the second definition is also a best approximation for the first one, but the converse is false.
The theory of continued fractions allows us to compute the best approximations of a real number: for the second definition, they are the convergents of its expression as a regular continued fraction. For the first definition, one has to consider also the semiconvergents.
For example, the constant e = 2.718281828459045235... has the (regular) continued fraction representation
A badly approximable number is an x for which there is a positive constant c such that for all rational p/q we have
It may be remarked that the preceding proof uses a variant of the pigeon hole principle: a non-negative integer that is not 0 is not smaller than 1. This apparently trivial remark is used in almost every proof of lower bounds for Diophantine approximations, even the most sophisticated ones.
In summary, a rational number is perfectly approximated by itself, but is badly approximated by any other rational number.
In the 1840s, Joseph Liouville obtained the first lower bound for the approximation of algebraic numbers: If x is an irrational algebraic number of degree n over the rational numbers, then there exists a constant c(x) > 0 such that
This result allowed him to produce the first proven example of a transcendental number, the Liouville constant
which does not satisfy Liouville's theorem, whichever degree n is chosen.
This link between Diophantine approximations and transcendental number theory continues to the present day. Many of the proof techniques are shared between the two areas.
Over more than a century, there were many efforts to improve Liouville's theorem: every improvement of the bound enables us to prove that more numbers are transcendental. The main improvements are due to Axel Thue (1909), Siegel (1921), Freeman Dyson (1947), and Klaus Roth (1955), leading finally to the Thue–Siegel–Roth theorem: If x is an irrational algebraic number and ε a (small) positive real number, then there exists a positive constant c(x, ε) such that
In some sense, this result is optimal, as the theorem would be false with ε=0. This is an immediate consequence of the upper bounds described below.
Subsequently, Wolfgang M. Schmidt generalized this to the case of simultaneous approximations, proving that: If x1, ..., xn are algebraic numbers such that 1, x1, ..., xn are linearly independent over the rational numbers and ε is any given positive real number, then there are only finitely many rational n-tuples (p1/q, ..., pn/q) such that
Again, this result is optimal in the sense that one may not remove ε from the exponent.
All preceding lower bounds are not effective, in the sense that the proofs do not provide any way to compute the constant implied in the statements. This means that one cannot use the results or their proofs to obtain bounds on the size of solutions of related Diophantine equations. However, these techniques and results can often be used to bound the number of solutions of such equations.
Nevertheless, a refinement of Baker's theorem by Feldman provides an effective bound: 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
However, as for every effective version of Baker's theorem, the constants d and 1/c are so large that this effective result cannot be used in practice.
This implies immediately that one cannot suppress the ε in the statement of Thue-Siegel-Roth theorem.
The equivalence may be read on the regular continued fraction representation, as shown by the following theorem of Serret:
Theorem: Two irrational numbers x and y are equivalent if and only there exist two positive integers h and k such that the regular continued fraction representations of x and y
Thus, except for a finite initial sequence, equivalent numbers have the same continued fraction representation.
Equivalent numbers are approximable to the same degree, in the sense that they have the same Markov constant.
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.
There are still simply-stated unsolved problems remaining in Diophantine approximation, for example the Littlewood conjecture and the lonely runner conjecture. It is also unknown if there are algebraic numbers with unbounded coefficients in their continued fraction expansion.
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.