View source for Diophantine approximation, metric theory of

The branch in number theory whose subject is the study of metric properties of numbers with special approximation properties (cf. [[Diophantine approximations|Diophantine approximations]]; [[Metric theory of numbers|Metric theory of numbers]]). One of the first theorems of the theory was Khinchin's theorem [[#References|[1]]], [[#References|[2]]] which, in its modern form [[#References|[3]]], may be stated as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d0325801.png" /> be a monotone decreasing function, defined for integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d0325802.png" />. Then the inequalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d0325803.png" /> have an infinite number of solutions in integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d0325804.png" /> for almost-all real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d0325805.png" /> if the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d0325806.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

diverges, and have only a finite number of solutions if this series converges (here and in what follows, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d0325807.png" /> is the distance from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d0325808.png" /> to the nearest integer, i.e.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d0325809.png" /></td> </tr></table>

where min is taken over all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258010.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258011.png" /> there exists an infinite number of rational approximations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258012.png" /> satisfying the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258013.png" /></td> </tr></table>

whereas the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258014.png" /></td> </tr></table>

has for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258015.png" /> an infinite number of solutions only for a set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258016.png" /> of measure zero.

The generalization of this theorem to simultaneous approximations [[#References|[3]]] is as follows. The system of inequalities

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

has a finite or infinite number of solutions for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258018.png" /> depending on whether the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

converges or diverges.

More extensive generalizations refer to systems of inequalities in several integer variables [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258020.png" />  "independent"  numbers and about the Lebesgue measure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258021.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258022.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258024.png" /> in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258025.png" /> for the convergence or divergence of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258026.png" /> to correspond to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258027.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258028.png" /> to be satisfied a finite or infinite number of times for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258029.png" />. A similar problem arises for the system of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258030.png" /> [[#References|[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 [[#References|[5]]]. The first one originated in the theory of transcendental numbers (Mahler's conjecture) and concerned simultaneous rational approximations to a system of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258031.png" /> for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258032.png" /> for any fixed natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258033.png" />. A recent result obtained on this subject runs as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258034.png" /> be a monotone decreasing function for which the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258035.png" /></td> </tr></table>

converges. Then the system of inequalities

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258036.png" /></td> </tr></table>

has only a finite number of solutions in integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258037.png" /> for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258038.png" /> [[#References|[7]]].

This theorem confirms that it is possible to approximate by rational numbers almost-all the points of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258039.png" />. Considerations of more general manifolds in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258040.png" /> will yield similar results.

If almost-all (in the sense of the measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258041.png" />) points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258042.png" /> of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258043.png" /> are such that the system (2) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258044.png" /> has a finite number of solutions in integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258045.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258047.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258048.png" /> is a curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258049.png" /> with non-zero curvature at almost-all its points, it is extremal [[#References|[8]]].

The method of trigonometric sums (cf. [[Trigonometric sums, method of|Trigonometric sums, method of]]; see also [[Vinogradov method|Vinogradov method]]) makes it possible to detect extremality of very general manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258050.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258051.png" />, under the condition that the topological dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258052.png" />. If, on the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258053.png" />, the extremal manifold cannot be quite general, and its structure should be fairly definite [[#References|[9]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. [A.Ya. Khinchin] Khintchine,   "Zur metrischen Theorie der diophantischen Approximationen"  ''Math. Z.'' , '''24'''  (1926)  pp. 706–714</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.Ya. [A.Ya. Khinchin] Khintchine,   "Kettenbrüche" , Teubner  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.W.S. Cassels,   "An introduction to diophantine approximation" , Cambridge Univ. Press  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.W.S. Cassels,   "Some metrical theorems in diophantine approximation I"  ''Proc. Cambridge Philos. Soc.'' , '''46''' :  2  (1950)  pp. 209–218</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.G. Sprindzhuk,   "Mahler's problem in metric number theory" , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.G. Sprindzhuk,   "New applications of analytic and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032580/d03258054.png" />-adic methods in Diophantine approximations" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''1''' , Gauthier-Villars  (1971)  pp. 505–509</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A. Baker,   "On a theorem of Sprindžuk"  ''Proc. Roy. Soc. Ser. A'' , '''292''' :  1428  (1966)  pp. 92–104</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  W. Schmidt,   "Metrische Sätze über simultane Approximation abhängiger Grössen"  ''Monatsh. Math.'' , '''68''' :  2  (1964)  pp. 154–166</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  V.G. Sprindzhuk,   "The metric theory of Diophantine approximations" , ''Current problems of analytic number theory'' , Minsk  (1974)  pp. 178–198  (In Russian)</TD></TR></table>