Difference between revisions of "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 | + | 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|Diophantine equations]]). |
If the (approximating) function under study | If the (approximating) function under study | ||
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<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/d032600/d03260079.png" /></td> </tr></table> | <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/d032600/d03260079.png" /></td> </tr></table> | ||
− | in integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260080.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260081.png" />), to find the rate of decrease of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260082.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260083.png" /> increases: In order for linear forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260084.png" /> to represent a | + | in integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260080.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260081.png" />), to find the rate of decrease of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260082.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260083.png" /> increases: In order for linear forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260084.png" /> to represent a "good" approximation to arbitrary numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260085.png" />, it is necessary and sufficient for these forms not to be a "good" approximation for the special sample of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260086.png" />. |
Problems in Diophantine approximations which are dissimilar at first sight sometimes turn out to be closely connected. For instance, Khinchin's transference principle [[#References|[1]]] relates the solvability of the equation | Problems in Diophantine approximations which are dissimilar at first sight sometimes turn out to be closely connected. For instance, Khinchin's transference principle [[#References|[1]]] relates the solvability of the equation | ||
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<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/d032600/d03260096.png" /></td> </tr></table> | <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/d032600/d03260096.png" /></td> </tr></table> | ||
− | In particular, the equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260098.png" /> are equivalent (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260099.png" /> then represent the | + | In particular, the equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260098.png" /> are equivalent (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260099.png" /> then represent the "worst" approximations, since equation (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600100.png" /> and equations (2) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600101.png" /> have an infinite number of solutions, whatever the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600102.png" />). Similar relations exist between the homogeneous and the inhomogeneous problems [[#References|[1]]], [[#References|[5]]], and not only for linear Diophantine approximations. If, for instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600103.png" /> are such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600104.png" /> for all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600105.png" />, |
<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/d032600/d032600106.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table> | <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/d032600/d032600106.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table> | ||
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<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/d032600/d032600112.png" /></td> </tr></table> | <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/d032600/d032600112.png" /></td> </tr></table> | ||
− | has an integer solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600113.png" /> subject to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600114.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600115.png" />. Moreover, the inequality (3) ensures a | + | has an integer solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600113.png" /> subject to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600114.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600115.png" />. Moreover, the inequality (3) ensures a "strong" uniform distribution of the fractional parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600116.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600117.png" />; the number of these fractions comprised in the system of intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600118.png" />, each one of which is located inside the unit interval, is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600119.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600120.png" /> is the length of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600121.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600122.png" /> is arbitrary. The validity of inequality (3) for all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600123.png" /> is equivalent to the validity of 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/d032600/d032600124.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table> | <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/d032600/d032600124.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table> | ||
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<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/d032600/d032600155.png" /></td> </tr></table> | <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/d032600/d032600155.png" /></td> </tr></table> | ||
− | depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600156.png" />. Since the magnitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600157.png" /> will be | + | depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600156.png" />. Since the magnitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600157.png" /> will be "smaller" only if at least one algebraic number of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600158.png" /> and height at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600159.png" /> is "close" to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600160.png" />, it follows that there is a connection between the estimation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600161.png" /> and the estimation of the approximation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600162.png" /> by algebraic numbers of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600163.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600164.png" />, where the minimum is taken over all algebraic numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600165.png" /> of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600166.png" /> and height at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600167.png" />, and let |
<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/d032600/d032600168.png" /></td> </tr></table> | <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/d032600/d032600168.png" /></td> </tr></table> | ||
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(see [[#References|[2]]]; [[Metric theory of numbers|Metric theory of numbers]]; [[Diophantine approximation, metric theory of|Diophantine approximation, metric theory of]]). | (see [[#References|[2]]]; [[Metric theory of numbers|Metric theory of numbers]]; [[Diophantine approximation, metric theory of|Diophantine approximation, metric theory of]]). | ||
− | The study of Diophantine equations by methods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600193.png" />-adic analysis stimulated the development of the theory of Diophantine approximations in the [[P-adic number|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600195.png" />-adic number]] fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600196.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600197.png" />. For instance, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600198.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600199.png" />-adic number. A consideration of approximations of zero (in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600200.png" />-adic metric) by the values of the integral linear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600201.png" /> yields rational approximations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600202.png" /> which, as in the case of real numbers, are closely connected with the expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600203.png" /> into a continued (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600204.png" />-adic) fraction [[#References|[10]]]. Analogues of the theorems of Dirichlet, Kronecker, Minkowski, etc., metric theorems, theorems on approximations by algebraic numbers, etc., are all valid [[#References|[2]]], [[#References|[6]]], [[#References|[8]]]. Diophantine inequalities in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600205.png" /> may be interpreted as congruences by a | + | The study of Diophantine equations by methods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600193.png" />-adic analysis stimulated the development of the theory of Diophantine approximations in the [[P-adic number|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600195.png" />-adic number]] fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600196.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600197.png" />. For instance, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600198.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600199.png" />-adic number. A consideration of approximations of zero (in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600200.png" />-adic metric) by the values of the integral linear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600201.png" /> yields rational approximations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600202.png" /> which, as in the case of real numbers, are closely connected with the expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600203.png" /> into a continued (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600204.png" />-adic) fraction [[#References|[10]]]. Analogues of the theorems of Dirichlet, Kronecker, Minkowski, etc., metric theorems, theorems on approximations by algebraic numbers, etc., are all valid [[#References|[2]]], [[#References|[6]]], [[#References|[8]]]. Diophantine inequalities in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600205.png" /> may be interpreted as congruences by a "high" degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600206.png" />, which makes it possible to obtain pure arithmetical theorems by an analytic method. A far-going development of Diophantine approximations in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600207.png" /> 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. [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600208.png" /> be an arbitrary algebraic field, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600209.png" /> be the ring of polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600210.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600211.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600212.png" /> be the field of power series of the form | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600208.png" /> be an arbitrary algebraic field, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600209.png" /> be the ring of polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600210.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600211.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600212.png" /> be the field of power series of the form | ||
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<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/d032600/d032600215.png" /></td> </tr></table> | <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/d032600/d032600215.png" /></td> </tr></table> | ||
− | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600216.png" /> is an arbitrary fixed number, in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600217.png" />. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600218.png" /> with the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600219.png" /> becomes a metric space. The study of | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600216.png" /> is an arbitrary fixed number, in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600217.png" />. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600218.png" /> with the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600219.png" /> becomes a metric space. The study of "Diophantine" approximations is carried out in the usual way, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600220.png" /> acting as the ring of integers: The approximating functions under consideration are functions, with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600221.png" />, of a finite number of variables with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600222.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600223.png" /> is replaced by the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600224.png" /> of series of the form |
<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/d032600/d032600225.png" /></td> </tr></table> | <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/d032600/d032600225.png" /></td> </tr></table> | ||
Line 115: | Line 115: | ||
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600228.png" />, explicit estimation of the accuracy of approximation, etc.). | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600228.png" />, 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 | + | 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|Diophantine approximation, metric theory of]]; [[Metric theory of numbers|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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600229.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600230.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600231.png" />, etc.) or else involves the construction of numbers with specified approximation properties (Liouville numbers, Mahler <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600232.png" />-numbers, etc.). The methods for solving such problems are specific and are often specially developed for a specific problem. |
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ. Press (1957) {{MR|0087708}} {{ZBL|0077.04801}} </TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top"> A. Baker, "On some Diophantine inequalities involving the exponential function" ''Canad. J. Math.'' , '''17''' (1965) pp. 616–626 {{MR|0177946}} {{ZBL|0147.30901}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> H. Davenport, W. Schmidt, "Approximation to real numbers by quadratic irrationals" ''Acta Arithm.'' , '''13''' (1967) pp. 169–176 {{MR|0219476}} {{ZBL|0155.09503}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.F. Koksma, "Diophantische Approximationen" , Springer (1936) {{MR|0344200}} {{MR|0004857}} {{MR|1545368}} {{ZBL|0012.39602}} {{ZBL|62.0173.01}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> E. Lutz, "Sur les approximations diophantiennes linéaires <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600233.png" />-adiques" , Hermann (1955) {{MR|69224}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> K. Mahler, "Ueber Beziehungen zwischen der Zahl $e$ und Liouvilleschen Zahlen" ''Math. Z.'' , '''31''' (1930) pp. 729–732</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> K. Mahler, "Ueber Diophantische Approximationen im Gebiete der <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600235.png" />-adischen Zahlen" ''Jahresber. Deutsch. Math-Verein.'' , '''44''' (1934) pp. 250–255</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> K. Mahler, "An analogue of Minkowski's theory of numbers in a field of series" ''Ann. of Math'' , '''42''' (1941) pp. 488–522</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> K. Mahler, "Lectures on Diophantine approximations" , '''1''' , Univ. Notre Dame (1961) {{MR|0142509}} {{ZBL|0158.29903}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> W. Schmidt, "Approximation to algebraic numbers" ''Enseign. Math. (2)'' , '''17''' : 3–4 (1971) pp. 187–253 {{MR|0327672}} {{ZBL|0226.10033}} {{ZBL|0217.03902}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> T. Schneider, "Einführung in die transzendenten Zahlen" , Springer (1957) {{MR|0086842}} {{ZBL|0077.04703}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> E. Wirsing, "Approximation mit algebraischen Zahlen beschränkten Grades" ''J. Reine Angew. Math.'' , '''206''' : 1–2 (1961) pp. 67–77 {{MR|0142510}} {{ZBL|0097.03503}} </TD></TR></table> |
Line 130: | Line 130: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Baker, "Transcendental number theory" , Cambridge Univ. Press (1975) {{MR|0422171}} {{ZBL|0297.10013}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W.M. Schmidt, "Linearformen. II" ''Math. Ann.'' , '''191''' (1971) pp. 1–20 {{MR|0308062}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W.M. Schmidt, "Diophantine Approximation" , ''Lect. notes in math.'' , '''785''' , Springer (1980) {{MR|0568710}} {{ZBL|0421.10019}} </TD></TR></table> |
Latest revision as of 18:32, 31 March 2017
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:
![]() |
(see [2]; Metric theory of numbers; Diophantine approximation, metric theory of).
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 ![]() |
[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 ![]() |
[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 |
Comments
The most important developments in Diophantine approximations are in the direction of transcendental number theory, irrational number theory and distribution modulo one.
Concerning the problem of representing numbers by norm forms one has the following, [a2]. Let be an algebraic number field and let
denote the norm map
. Let
be a module in
, i.e., a finite dimensional
-module
(also called an (incomplete) lattice). One speaks of a full module if
. Then a necessary and sufficient condition for there to exist an integer
such that the equation
has infinitely many solutions
in
is that
be a full module in some subfield of
that is neither the rational nor an imaginary quadratic field.
Let be a basis for a module
. Consider the linear form
. Let
run through the imbeddings of
into the complex numbers
. Let
. The product
is a homogeneous form of degree
over
. Such forms are called norm forms, and solving
is of course the same as representing
by the form
(with entries from
).
References
[a1] | A. Baker, "Transcendental number theory" , Cambridge Univ. Press (1975) MR0422171 Zbl 0297.10013 |
[a2] | W.M. Schmidt, "Linearformen. II" Math. Ann. , 191 (1971) pp. 1–20 MR0308062 |
[a3] | W.M. Schmidt, "Diophantine Approximation" , Lect. notes in math. , 785 , Springer (1980) MR0568710 Zbl 0421.10019 |
Diophantine approximations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diophantine_approximations&oldid=11927