Diophantine Approximations

a part of the theory of numbers that studies approximations of real numbers by rational numbers or, in a broader context, problems involved in finding integral solutions of linear and nonlinear inequalities or systems of inequalities with real coefficients. Diophantine approximations are named after the ancient Greek mathematician Diophantus, who worked on the problem of finding integral solutions of algebraic equations (Diophantine equations). The methods of the theory of Diophantine approximations are based on the application of continued fractions, Farey sequences, and the Dirichlet principle.

The problem of approximating a number by rational fractions is solved with the aid of all three of these methods and particularly with the use of continued fractions. The approximation of a real number α by convergents p_k/q_k of the expansion of α into a continued fraction is characterized by the inequality │ α − p_k/q_k │ < 1/q_k². On the other hand, if an irreducible fraction a/b satisfies the inequality │ α − a/b │ < 1/2b², then it is a convergent in the expansion of α into a continued fraction. Fundamental work on the approximation of real numbers α by rational fractions has been done by A. A. Markov (Senior). There are many extensions of the problem of approximating a number by rational fractions. Among them, the primary problem is that of studying expressions of the type xθ − y − α, where θ and α are certain real numbers and x and y assume integral values (the so-called nonhomogeneous one-dimensional problem). The first results in solving this problem were achieved by P. L. Chebyshev. A well-known theorem on approximate integral solutions of systems of linear equations (multidimensional problems of Diophantine approximations), is the following theorem of L. Kronecker: If α₁, …, α_n are real numbers for which the equality a₁α₁ + … + a_nα_n = 0 with integral a₁, …, a_n holds only for a₁ = … = a_n = 0 and β₁, …, β_n are certain real numbers, then for any ε > 0 it is possible to find a number t and also integers x₁ …, x_n, such that the inequalities │ t α_k − β_k − x_k │ < ε for k = 1, 2, … , n. Dirichlet’ s principle is very useful in solving multidimensional problems of Diophantine approximations. Methods based on Dirichlet’s principle enabled A. Ia. Khinchin and other mathematicians to develop a systematic theory of multidimensional Diophantine approximations. An important aspect of the theory of Diophantine approximations is its connection with geometry based on the fact that a system of linear forms with real coefficients can be represented as a lattice in an n-dimensional arithmetic space. In the late 19th century H. Minkowski proved a number of geometric theorems applicable to the theory of Diophantine approximations.

I. M. Vinogradov obtained remarkable results in problems of nonlinear Diophantine approximations. The methods devised by him occupy a central position in this area of number theory. One of the most important problems of the theory of Diophantine approximations is that of approximating algebraic numbers by rational numbers.

The theory of transcendental numbers, in which estimates are found for norms of linear forms and polynomials in one and several numbers with integral coefficients, is related to Diophantine approximations. The theory of Diophantine approximations is closely related to the solution of Diophantine equations and to various problems of analytic number theory.