Diophantine approximation, metric theory of

2020 Mathematics Subject Classification: Primary: 11J83 [MSN][ZBL]

The branch in number theory whose subject is the study of metric properties of numbers with special approximation properties (cf. Diophantine approximations; Metric theory of numbers). One of the first theorems of the theory was Khinchin's theorem [1], [2] which, in its modern form [3], may be stated as follows. Let $\phi(q)$ be a monotone decreasing function, defined for integers $q \ge 1$. Then the inequalities $\Vert \alpha q \Vert < \phi(q)$ have an infinite number of solutions in integers $q \ge 1$ for almost-all real numbers $\alpha$ if the series $$ \sum_{q=1}^\infty \phi(q) $$ diverges, and have only a finite number of solutions if this series converges (here and in what follows, $\Vert \alpha \Vert$ is the distance from $\alpha$ to the nearest integer, i.e. $$ \Vert x \Vert = \min_n |x-n|\,, $$ where $\min$ is taken over all integers $n \in \mathbf{Z}$; the term "almost-all" refers to Lebesgue measure in the respective space). The theorem describes the accuracy of the approximation of almost-all real numbers by rational fractions. For example, for almost-all $\alpha$ there exists an infinite number of rational approximations $p/q$ satisfying the inequality $$ \left\vert{ \alpha - \frac{p}{q} }\right\vert < \frac{1}{q^2 \log q} $$ whereas the inequality $$ \left\vert{ \alpha - \frac{p}{q} }\right\vert < \frac{1}{q^2 (\log q)^{1+\epsilon}} $$ has for any $\epsilon>0$ an infinite number of solutions only for a set of numbers $\alpha$ of measure zero.

The generalization of this theorem to simultaneous approximations [3] is as follows. The system of inequalities $$ \max(\Vert \alpha_1 q \Vert, \ldots, \Vert \alpha_n q \Vert) < \phi(q) $$ has a finite or infinite number of solutions for almost-all $(\alpha_1,\ldots,\alpha_n) \in \mathbf{R}^n$ depending on whether the series $$ \sum_{q=1}^\infty \phi(q)^n $$ converges or diverges.

More extensive generalizations refer to systems of inequalities in several integer variables [5].

A distinguishing feature of Khinchin's theorem and its many generalizations is the fact that the property of "convergence-divergence" of series of the types (1), (3) serves as a criterion of the corresponding order of the approximation applying to a set of numbers of measure zero or to almost-all numbers. It is a kind of "zero-one" law for the metric theory of Diophantine approximations. Another characteristic of these generalizations is the fact that the metric property of the numbers involved refers to a measure defined throughout the space containing the numbers which participate in the approximation, and that the measure of the space is defined as the product of the measures in the coordinate spaces. For instance, in the case of system (2) one speaks about an approximation of $n$ "independent" numbers and about the Lebesgue measure in $\mathbf{R}^n = \mathbf{R} \times\cdots\times \mathbf{R}$ ($n$ times). In this connection this part of the theory received the name of metric theory of Diophantine approximations of independent variables. It has been fairly thoroughly developed, but a number of unsolved problems still (1988) remain. One such problem concerns the conditions which must be imposed on a sequence of measurable sets $A(q)$, $q=1,2,\ldots$ in the interval $[0,1]$ for the convergence or divergence of the series $\sum_q |A(q)|$ to correspond to the condition $\alpha q \in A(q) \pmod{1}$ to be satisfied a finite or infinite number of times for almost-all $\alpha$. A similar problem arises for the system of numbers $(\alpha_1 q,\ldots,\alpha_n q)$ [4].

The metric theory of Diophantine approximations of dependent variables, which is of a later date, immediately gave rise to several fundamental and characteristic problems [5]. The first one originated in the theory of transcendental numbers (Mahler's conjecture) and concerned simultaneous rational approximations to a system of numbers $t,\ldots,t^n$ for almost-all $t$ for any fixed natural number $n$. A recent result obtained on this subject runs as follows. Let $\phi(q)>0$ be a monotone decreasing function for which the series $$ \sum_{q=1}^\infty \frac{\phi(q)}{q} $$ converges. Then the system of inequalities $$ \max(\Vert t q \Vert, \ldots, \Vert t^n q \Vert) < \frac{\phi(q)^n}{q^n} $$ has only a finite number of solutions in integers $q \ge 1$ for almost-all $t$ [7].

This theorem confirms that it is possible to approximate by rational numbers almost-all the points of a curve $\Gamma \subset \mathbf{R}^n$. Considerations of more general manifolds in $\mathbf{R}^n$ will yield similar results.

If almost-all (in the sense of the measure on $\Gamma$) points $(\alpha_1,\ldots,\alpha_n)$ of the manifold $\Gamma$ are such that the system (2) with $\phi(q) = q^{-1/n-\epsilon}$ has a finite number of solutions in integers $q\ge1$ for any $\epsilon>0$, then $\Gamma$ is said to be extremal, i.e. almost-all points permit only the worst simultaneous approximation by rational numbers. Schmidt's theorem says that if $\Gamma$ is a curve in $\mathbf{R}^2$ with non-zero curvature at almost-all its points, it is extremal [8].

The method of trigonometric sums (cf. Trigonometric sums, method of; see also Vinogradov method) makes it possible to detect extremality of very general manifolds $\Gamma$ in $\mathbf{R}^n$, under the condition that the topological dimension $\dim\Gamma \ge n/2$. If, on the other hand, $\dim\Gamma < n/2$, the extremal manifold cannot be quite general, and its structure should be fairly definite [9].

References

[1]	A. [A.Ya. Khinchin] Khintchine, "Zur metrischen Theorie der diophantischen Approximationen" Math. Z. , 24 (1926) pp. 706–714
[2]	A.Ya. [A.Ya. Khinchin] Khintchine, "Kettenbrüche" , Teubner (1956) (Translated from Russian)
[3]	J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ. Press (1957)
[4]	J.W.S. Cassels, "Some metrical theorems in diophantine approximation I" Proc. Cambridge Philos. Soc. , 46 : 2 (1950) pp. 209–218
[5]	V.G. Sprindzhuk, "Mahler's problem in metric number theory" , Amer. Math. Soc. (1969) (Translated from Russian)
[6]	V.G. Sprindzhuk, "New applications of analytic and $p$-adic methods in Diophantine approximations" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 1 , Gauthier-Villars (1971) pp. 505–509
[7]	A. Baker, "On a theorem of Sprindžuk" Proc. Roy. Soc. Ser. A , 292 : 1428 (1966) pp. 92–104
[8]	W. Schmidt, "Metrische Sätze über simultane Approximation abhängiger Grössen" Monatsh. Math. , 68 : 2 (1964) pp. 154–166
[9]	V.G. Sprindzhuk, "The method of trigonometric sums in the metric theory of diophantine approximations of dependent quantities" Proc. Steklov Inst. Math. , 128 : 2 (1972) pp. 251–270 Trudy Mat. Inst. Steklov. , 128 : 2 (1972) pp. 212–228
[10]	V.G. Sprindzhuk, "The metric theory of Diophantine approximations" , Current problems of analytic number theory , Minsk (1974) pp. 178–198 (In Russian)