Diophantine approximations

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The branch of number theory whose subject is the approximation of zero by values of functions of a finite number of integer arguments. The original problems of Diophantine approximations concerned rational approximations to real numbers, but the development of the theory gave rise to problems in which certain real functions must be assigned "small" values if the values of the arguments are integers. Accordingly, Diophantine approximations are closely connected with solving inequalities in integers — Diophantine inequalities — and also with solving equations in integers (cf. Diophantine equations).

If the (approximating) function under study

is linear with respect to the integer arguments , then the Diophantine approximations with the function are said to be linear; otherwise they are called non-linear. If is a homogeneous polynomial in , the Diophantine approximations with the function are said to be homogeneous. Several functions with at least one common integer argument may be studied at the same time. In such a case the Diophantine approximations are called simultaneous. Simultaneous Diophantine approximations may be linear or non-linear, homogeneous or inhomogeneous in the above sense.

Numerical values of may be considered as being close to zero not only if

for a given , but also if

(one-sided approximations). The function may depend on parameters which continuously vary in some domain; these are parametric Diophantine approximations. Finally, the domain of definition and the range of values of the approximating functions may be subsets not only of a Euclidean space, but also of altogether different topological spaces (see below: Diophantine approximations in -adic number fields and Diophantine approximations in the field of power series).

The oldest ( "simplest" ) problem in Diophantine approximations are approximations of zero by a linear form , where is a given real number and and are variable integers (linear homogeneous Diophantine approximations), i.e. the problem of rational approximations to . For special () this problem had been considered even in Antiquity (Archimedes, Diophantus, Euclid), while its close connection with the theory of continued fractions (cf. Continued fraction) was completely clarified by L. Euler and J.L. Lagrange. In particular, if , are such that

where the minimum is taken over all integers in some arbitrary interval and over all integer values , the fraction is a convergent fraction of the expansion of into a continued fraction. If the incomplete partial fractions of into a continued fraction are bounded, then there exists a with the condition for all integers . This is true, for example, for quadratic irrationalities (cf. Quadratic irrationality), since then the expansion into a continued fraction is periodic. On the other hand, for any irrational number the inequality has an infinite number of integer solutions , and if , the constant cannot be replaced by a smaller number. The study of A.A. Markov on the minima of indefinite binary quadratic forms (cf. Binary quadratic form) made it possible to extend this last statement: If is not equivalent (in the sense of the theory of continued fractions) to , then the inequality has an infinite number of solutions; the constant cannot be improved upon if is equivalent to ; if is not equivalent either to or to , the inequality has an infinite number of solutions, etc. [1]. The constants decrease monotonically and have limit .

The simplest example of linear inhomogeneous Diophantine approximations are approximations of zero by a linear inhomogeneous polynomial , where are real numbers and are integer variables. It was shown by P.L. Chebyshev that for any irrational number and any the inequality has an infinite number of solutions in integers , . In this case 2 is not the best constant: It was proved by H. Minkowski that if , where are integers, the constant 2 can be replaced by , the latter being the optimal constant. This statement is a corollary of the simplest case of a hypothesis on the product of inhomogeneous linear forms proved by H. Minkowski himself (cf. Minkowski hypothesis).

More complex problems of the general theory of Diophantine approximations concern the approximation of functions of a large number of integer arguments (cf. Dirichlet theorem; Minkowski theorem; Kronecker theorem). It is convenient to introduce the function , where the minimum is taken over all integers (the distance between and the nearest integer). For instance, the above-mentioned linear polynomials and may be replaced by and for integers . It follows from Dirichlet's theorem that for all real there exists an infinite number of solutions of the system of inequalities

in integers . Here, 1 may be replaced by a smaller number (e.g. ), but the optimal constant is unknown for any (1988). It cannot be an arbitrary number, as is shown by the example of numbers which form a basis of a real algebraic field [1]. If are linearly independent over the field of rational numbers, then for any and any there exists an infinite number of solutions of the system of inequalities

in integers (Kronecker's theorem). An important feature of this theorem on simultaneous inhomogeneous Diophantine approximations consists in the fact that it is not possible, in principle (without special information on homogeneous approximations to ), to find the rate of decrease of as increases: In order for linear forms to represent a "good" approximation to arbitrary numbers , it is necessary and sufficient for these forms not to be a "good" approximation for the special sample of numbers .

Problems in Diophantine approximations which are dissimilar at first sight sometimes turn out to be closely connected. For instance, Khinchin's transference principle [1] relates the solvability of the equation

 (1)

in integers to that of the system

 (2)

in integers , and vice versa: If and are, respectively, the least upper bounds of those and for which (1) and (2) have an infinite number of solutions, then

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 [1], [5], and not only for linear Diophantine approximations. If, for instance, are such that for any for all integers ,

 (3)

where depends only on and , then, for any real numbers and any , the system of inequalities

has an integer solution subject to the condition if . 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

 (4)

for all integers for any , where depends only on and .

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 1844 (cf. Liouville number), has not yet been conclusively solved (cf. Thue–Siegel–Roth theorem; Diophantine approximation, problems of effective). It has been shown [11] 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). W. Schmidt [11] used such ideas to obtain a complete solution of the problem of representing numbers by norm forms, basing himself on his approximation theorem. 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. 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 [12]. For instance, it has been shown by K. Mahler [7], [12] that

where is an absolute constant and . A. Baker [3] used another method to demonstrate (4) for various non-zero rational powers of with , where

depends only on . 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

E. Wirsing [13] 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

where is the height of , and is arbitrary. Wirsing conjectured that for all transcendental and all . In addition to the self-evident case , this conjecture has been demonstrated for [4]. 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 [10]. Analogues of the theorems of Dirichlet, Kronecker, Minkowski, etc., metric theorems, theorems on approximations by algebraic numbers, etc., are all valid [2], [6], [8]. 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. [10].

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

One introduces a non-Archimedean valuation,

where is an arbitrary fixed number, in the field . 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

the results are analogous to approximations in a -adic number field [2], [9].

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. The global approach involves the study of general laws of approximation, which apply to all numbers or to all numbers with "rare" exceptions. This is the case of the Dirichlet theorem on homogeneous approximations, Kronecker's theorem on inhomogeneous approximations, general theorems on the approximation of numbers by algebraic numbers, classifications of numbers by their approximation properties, etc. The corresponding methods are "global" (continued fractions, etc.). The metric approach involves the description of the approximation properties of numbers on the base of concepts of measure theory (cf. Diophantine approximation, metric theory of; Metric theory of numbers). The results thus obtained do not apply to all, but to almost-all (in the sense of a definite measure) numbers in the sets under consideration or else are described with the aid of some metric characteristic (the Hausdorff dimension, the capacity, etc.). The methods used are closely connected with measure theory, probability theory and related disciplines. The individual approach concerns the approximation properties of special numbers (algebraic numbers, , , , etc.) or else involves the construction of numbers with specified approximation properties (Liouville numbers, Mahler -numbers, etc.). The methods for solving such problems are specific and are often specially developed for a specific problem.

References

 [1] J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ. Press (1957) MR0087708 Zbl 0077.04801 [2] V.G. Sprindzhuk, "Mahler's problem in metric number theory" , Amer. Math. Soc. (1969) (Translated from Russian) [3] A. Baker, "On some Diophantine inequalities involving the exponential function" Canad. J. Math. , 17 (1965) pp. 616–626 MR0177946 Zbl 0147.30901 [4] H. Davenport, W. Schmidt, "Approximation to real numbers by quadratic irrationals" Acta Arithm. , 13 (1967) pp. 169–176 MR0219476 Zbl 0155.09503 [5] J.F. Koksma, "Diophantische Approximationen" , Springer (1936) MR0344200 MR0004857 MR1545368 Zbl 0012.39602 Zbl 62.0173.01 [6] E. Lutz, "Sur les approximations diophantiennes linéaires -adiques" , Hermann (1955) MR69224 [7] K. Mahler, "Ueber Beziehungen zwischen der Zahl \$e\$ und Liouvilleschen Zahlen" Math. Z. , 31 (1930) pp. 729–732 [8] K. Mahler, "Ueber Diophantische Approximationen im Gebiete der -adischen Zahlen" Jahresber. Deutsch. Math-Verein. , 44 (1934) pp. 250–255 [9] K. Mahler, "An analogue of Minkowski's theory of numbers in a field of series" Ann. of Math , 42 (1941) pp. 488–522 [10] K. Mahler, "Lectures on Diophantine approximations" , 1 , Univ. Notre Dame (1961) MR0142509 Zbl 0158.29903 [11] W. Schmidt, "Approximation to algebraic numbers" Enseign. Math. (2) , 17 : 3–4 (1971) pp. 187–253 MR0327672 Zbl 0226.10033 Zbl 0217.03902 [12] T. Schneider, "Einführung in die transzendenten Zahlen" , Springer (1957) MR0086842 Zbl 0077.04703 [13] E. Wirsing, "Approximation mit algebraischen Zahlen beschränkten Grades" J. Reine Angew. Math. , 206 : 1–2 (1961) pp. 67–77 MR0142510 Zbl 0097.03503