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Handbook of Dynamical Systems B. Hasselblatt , A. General Formalism. An Introduction to Dynamical Systems D. For instance, Khinchin's transference principle  relates the solvability of the equation. In particular, the equalities and are equivalent the then represent the "worst" approximations, since equation 1 with and equations 2 with have an infinite number of solutions, whatever the values of.
Similar relations exist between the homogeneous and the inhomogeneous problems  ,  , and not only for linear Diophantine approximations.
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If, for instance, are such that for any for all integers ,. Moreover, the inequality 3 ensures a "strong" uniform distribution of the fractional parts , where ; the number of these fractions comprised in the system of intervals , each one of which is located inside the unit interval, is , where is the length of the interval and is arbitrary. The validity of inequality 3 for all integers is equivalent to the validity of the inequality. The proof of the solvability or non-solvability of Diophantine inequalities whose parameters are determined by arithmetical or analytical conditions is often a very complex task.
Thus, the problem of approximating algebraic numbers by rational numbers, which has been systematically studied ever since the Liouville inequality was demonstrated in cf. Liouville number , has not yet been conclusively solved cf. Thue—Siegel—Roth theorem ; Diophantine approximation, problems of effective. It has been shown  that for algebraic numbers which are together with 1 linearly independent over the field of rational numbers, the inequalities 3 and 4 are valid for any.
It follows that the system of inequalities 1 for any and the system of inequalities 2 for any have only a finite number of solutions. There is a close connection between such theorems and Diophantine approximations to algebraic numbers and the representation of integers by incomplete norm forms. In particular, the problem of bounds for the solutions of Thue's Diophantine equation , for a given integral irreducible binary form of degree at least three and a variable integer , is equivalent to the study of rational approximations to a root of the polynomial.
In this way A. Thue showed that the number of solutions of the equation is finite, having previously obtained a non-trivial estimate for rational approximations to. This approach, generalized and developed by C. Siegel, led him to the theorem that the number of integral points on algebraic curves of genus higher than zero is finite cf. Diophantine geometry. Schmidt  used such ideas to obtain a complete solution of the problem of representing numbers by norm forms, basing himself on his approximation theorem.
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In certain cases the connections between the theory of Diophantine equations and that of Diophantine approximations of numbers may play a main role in proofs on the existence of solutions in the Waring problem and in the method of Hardy—Littlewood—Vinogradov. Diophantine approximations to special numbers, given as the values of transcendental functions at rational or algebraic points, are studied by methods of the theory of transcendental numbers cf. Transcendental number.
As a rule, if it can be proved that some number is irrational or transcendental, it is also possible to estimate its approximation by rational or algebraic numbers.
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In the case of a transcendental , the magnitude , where the minimum is taken over all non-zero integer polynomials of degree at most and height at most , is called the measure of transcendency of the number. An estimate from below of , mainly for a fixed and a variable , forms the subject of many theorems in transcendental number theory .
For instance, it has been shown by K. Mahler  ,  that.
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Baker  used another method to demonstrate 4 for various non-zero rational powers of with , where. Since the magnitude will be "smaller" only if at least one algebraic number of degree at most and height at most is "close" to , it follows that there is a connection between the estimation of and the estimation of the approximation to by algebraic numbers of degree at most. Let , where the minimum is taken over all algebraic numbers of degree at most and height at most , and let.
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Wirsing  found relations between and if is a real number:. In particular, if , then , and since for all transcendental , it follows that. This means that for any transcendental there exists an infinite number of algebraic 's of degree at most satisfying the inequality. Wirsing conjectured that for all transcendental and all. In addition to the self-evident case , this conjecture has been demonstrated for .
It is also known that for almost-all in the sense of Lebesgue measure real the following equalities are valid:. The study of Diophantine equations by methods of -adic analysis stimulated the development of the theory of Diophantine approximations in the -adic number fields , the structure of which is parallel in many respects to the theory of Diophantine approximations in the field of real numbers, but taking into account the non-Archimedean topology of. For instance, let be a -adic number. A consideration of approximations of zero in the -adic metric by the values of the integral linear form yields rational approximations of which, as in the case of real numbers, are closely connected with the expansion of into a continued -adic fraction .
Analogues of the theorems of Dirichlet, Kronecker, Minkowski, etc. Diophantine inequalities in may be interpreted as congruences by a "high" degree of , which makes it possible to obtain pure arithmetical theorems by an analytic method. A far-going development of Diophantine approximations in the field and its finite extensions makes it possible to use the Thue—Siegel—Roth method to demonstrate theorems on the arithmetical structure of numbers representable by binary forms, on estimates of the fractional parts of powers of rational numbers, etc.
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Since the expansion of functions into continued fractions is similar to the expansion of numbers into continued fractions, a further analogy arises naturally — approximations of a function by rational functions in the metric of a field of power series. This approach has been considerably developed and leads to the theory of Diophantine approximations in a field of power series. Let be an arbitrary algebraic field, let be the ring of polynomials in over and let be the field of power series of the form.
The field with the norm becomes a metric space. The study of "Diophantine" approximations is carried out in the usual way, with acting as the ring of integers: The approximating functions under consideration are functions, with values in , of a finite number of variables with values in , while the estimation is carried out with respect to the norm introduced.
There is a certain similarity between results obtained in this manner and the case of Diophantine approximations in the field of real numbers, but if is replaced by the field of series of the form. Diophantine approximations in a field of power series form a more concrete basis of certain analytic methods in the theory of transcendental numbers specialization of , explicit estimation of the accuracy of approximation, etc.
Three different approaches in the development of the theory of Diophantine approximations may be distinguished: global, metric and individual.