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| In [[number theory]], the field of '''Diophantine approximation''', named after [[Diophantus of Alexandria]], deals with the approximation of [[real number]]s by [[rational number]]s.
| | I am 34 years old and my name is Katrice Houser. I life in Hamburg Alsterdorf (Germany).<br><br>Feel free to visit my blog; [http://safedietplans.com/bmr-calculator bmr calculator] |
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| The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number ''a''/''b'' is a "good" approximation of a real number ''α'' if the absolute value of the difference between ''a''/''b'' and ''α'' may not decrease if ''a''/''b'' is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of [[continued fraction]]s.
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| Knowing the "best" approximations of a given number, the main problem of the field is to find sharp upper and lower bounds of the above difference, expressed as a function of the denominator.
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| It appears that these bounds depend on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number is larger than the lower bound for [[algebraic number]]s, which is itself larger than the lower bound for all real numbers. Thus a real number that may be better approximated than the bound for algebraic numbers is certainly a [[transcendental number]]. This allowed [[Joseph Liouville|Liouville]], in 1844 to produce the first explicit transcendental number. Later, the proofs that [[Pi|{{pi}}]] and ''[[e (mathematical constant)|e]]'' are transcendental were obtained with a similar method.
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| Thus Diophantine approximations and [[transcendence theory]] are very close areas that share many theorems and methods. Diophantine approximations have also important applications in the study of [[Diophantine equation]]s.
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| == Best Diophantine approximations of a real number ==
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| {{main|Continued fraction#Best rational approximations}}
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| Given a real number {{math|''α''}}, there are two ways to define a best Diophantine approximation of {{math|''α''}}. For the first definition,<ref name="Khinchin 1997 p.21">Khinchin (1997) p.21</ref> the rational number {{math|''p''/''q''}} is a ''best Diophantine approximation'' of {{math|''α''}} if
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| :<math>\left|\alpha -\frac{p}{q}\right | < \left|\alpha -\frac{p'}{q'}\right |,</math>
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| for every rational number {{math|''p'''/''q' ''}} different of {{math|''p''/''q''}} such that {{math|0 < ''q' ''≤ ''q''}}.
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| For the second definition,<ref>Cassels (1957) p.2</ref><ref name=Lang9>Lang (1995) p.9</ref> the above inequality is replaced by
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| :<math>\left|q\alpha -p\right| < \left|q^\prime\alpha - p^\prime\right|.</math>
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| A best approximation for the second definition is also a best approximation for the first one, but the converse is false.<ref name=Khinchin24>Khinchin (1997) p.24</ref>
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| The theory of [[continued fraction]]s allows us to compute the best approximations of a real number: for the second definition, they are the [[convergent (continued fraction)|convergents]] of its expression as a regular continued fraction.<ref name=Lang9/><ref name=Khinchin24/><ref>Cassels (1957) pp.5–8</ref> For the first definition, one has to consider also the [[Continued fraction#Semiconvergents|semiconvergents]].<ref name="Khinchin 1997 p.21"/>
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| For example, the constant ''e'' = 2.718281828459045235... has the (regular) continued fraction representation
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| :<math>[2;1,2,1,1,4,1,1,6,1,1,8,1,\ldots\;].</math>
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| Its best approximations for the second definition are
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| :<math> 3, \tfrac{8}{3}, \tfrac{11}{4}, \tfrac{19}{7}, \tfrac{87}{32}, \ldots\, ,</math>
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| while, for the first definition, they are
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| :<math>3, \tfrac{5}{2}, \tfrac{8}{3}, \tfrac{11}{4}, \tfrac{19}{7}, \tfrac{30}{11},
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| \tfrac{49}{18}, \tfrac{68}{25}, \tfrac{87}{32}, \tfrac{106}{39}, \ldots\, .</math>
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| ==Measure of the accuracy of approximations ==
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| The obvious measure of the accuracy of a Diophantine approximation of a real number {{math|''α''}} by a rational number {{math|''p''/''q''}} is <math>\left|\alpha-\frac{p}{q}\right|.</math> However, this quantity may always be made arbitrarily small by increasing the absolute values of {{math|''p''}} and {{math|''q''}}; thus the accuracy of the approximation is usually estimated by comparing this quantity to some function {{math|''φ''}} of the denominator {{math|''q''}}, typically a negative power of it.
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| For such a comparison, one may want upper bounds or lower bounds of the accuracy. A lower bound is typically described by a theorem like "for every element {{math|''α''}} of some subset of the real numbers and every rational number {{math|''p''/''q''}}, we have <math>\left|\alpha-\frac{p}{q}\right|>\phi(q)</math> ". In some case, "every rational number" may be replaced by "all rational numbers except a finite number of them", which amounts to multiplying {{math|''φ''}} by some constant depending on {{math|''α''}}.
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| For upper bounds, one has to take into accounts that not all the "best" Diophantine approximations provided by the convergents may have the desired accuracy. Therefore the theorems take the form "for every element {{math|''α''}} of some subset of the real numbers, there are infinitely many rational numbers {{math|''p''/''q''}} such that <math>\left|\alpha-\frac{p}{q}\right|<\phi(q)</math> ".
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| ===Badly approximable numbers===
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| A '''badly approximable number''' is an ''x'' for which there is a positive constant ''c'' such that for all rational ''p''/''q'' we have
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| :<math>\left|{ x - \frac{p}{q} }\right| > \frac{c}{q^2} \ . </math>
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| The badly approximable numbers are precisely those with bounded [[convergent (continued fraction)|partial quotients]].<ref name=Bug245>Bugeaud (2012) p.245</ref>
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| == Lower bounds for Diophantine approximations ==
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| === Approximation of a rational by other rationals ===
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| A rational number <math>\alpha =\frac{a}{b}</math> may be obviously and perfectly approximated by <math>\tfrac{p_i}{q_i} = \tfrac{i\,a}{i \,b}</math> for every positive integer ''i''.
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| If <math>\tfrac{p}{q} \not= \alpha = \tfrac{a}{b}\,,</math> we have
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| :<math> \left|\frac{a}{b} - \frac{p}{q}\right| = \left|\frac{aq-bp}{bq}\right| \ge \frac{1}{bq},</math>
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| because <math>|aq-bp|</math> is a positive integer and is thus not lower than 1. Thus the accuracy of the approximation is bad relative to irrational numbers (see next sections).
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| It may be remarked that the preceding proof uses a variant of the [[pigeon hole principle]]: a non-negative integer that is not 0 is not smaller than 1. This apparently trivial remark is used in almost every proof of lower bounds for Diophantine approximations, even the most sophisticated ones.
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| In summary, a rational number is perfectly approximated by itself, but is badly approximated by any other rational number.
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| === Approximation of algebraic numbers, Liouville's result ===
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| {{main|Liouville number}}
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| In the 1840s, [[Joseph Liouville]] obtained the first lower bound for the approximation of [[algebraic number]]s: If ''x'' is an irrational algebraic number of degree ''n'' over the rational numbers, then there exists a constant {{nowrap|''c''(''x'') > 0}} such that
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| :<math> \left| x- \frac{p}{q} \right| > \frac{c(x)}{q^{n}}</math>
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| holds for every integers ''p'' and ''q'' where {{nowrap|''q'' > 0}}.
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| This result allowed him to produce the first proven example of a transcendental number, the [[Liouville constant]]
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| :<math>
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| \sum_{j=1}^\infty 10^{-j!} = 0.110001000000000000000001000\ldots\,,
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| </math>
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| which does not satisfy Liouville's theorem, whichever degree ''n'' is chosen.
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| This link between Diophantine approximations and [[transcendental number|transcendence theory]] continues to the present-day. Many of the proof techniques are shared between the two areas.
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| === Approximation of algebraic numbers, Thue–Siegel–Roth theorem ===
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| {{main|Thue–Siegel–Roth theorem}}
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| During more than a century, there were many efforts to improve Liouville's theorem: every improvement of the bound allows to prove that more numbers are transcendental. The main improvements are due to [[Axel Thue]] (1909), [[Carl Ludwig Siegel]] (1921), [[Freeman Dyson]] (1947) and [[Klaus Roth]] (1955), leading finally to the so-called Thue–Siegel–Roth theorem: If {{math|''x''}} is an irrational algebraic number and {{math|''ε''}} a (small) positive real number, then there exists a positive constant {{math|''c''(''x'', ''ε'')}} such that
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| :<math>
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| \left| x- \frac{p}{q} \right|>\frac{c(x, \varepsilon)}{q^{2+\varepsilon}}
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| </math> | |
| holds for every integers {{math|''p''}} and {{math|''q''}} such that {{math|''q'' > 0}}.
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| In some sense, this result is optimal, as the theorem would be false with ''ε''=0. This is an immediate consequence of the upper bounds described below.
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| === Simultaneous approximations of algebraic numbers ===
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| {{main|Subspace theorem}}
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| Subsequently, [[Wolfgang M. Schmidt]] generalized this to the case of simultaneous approximations, proving that: If {{math|''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>}} are algebraic numbers such that {{math|1, ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>}} are linearly independent over the rational numbers and {{math|''ε''}} is any given positive real number, then there are only finitely many rational {{math|''n''}}-tuples {{math|(''p''<sub>1</sub>/''q'', ..., ''p''<sub>''n''</sub>/''q'')}} such that
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| :<math>|x_i-p_i/q|<q^{-(1+1/n+\varepsilon)},\quad i=1,\ldots,n.</math>
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| Again, this result is optimal in the sense that one may not remove {{math|''ε''}} from the exponent.
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| === Effective bounds ===
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| All preceding lower bounds are not [[effective results in number theory|effective]], in the sense that the proofs do not provide any way to compute the constant implied in the statements. This means that one cannot use the results or their proofs to obtain bounds on the size of solutions of related Diophantine equations. However, these techniques and results can often be used to bound the number of solutions of such equations.
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| Nevertheless a refinement of [[Baker's theorem]] by Feldman provides an effective bound: if ''x'' is an algebraic number of degree ''n'' over the rational numbers, then there exist effectively computable constants ''c''(''x'') > 0 and 0 < ''d''(''x'') < ''n'' such that
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| :<math>\left| x- \frac{p}{q} \right|>\frac{c(x)}{|q|^{d(x)}} </math>
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| holds for all rational integers.
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| However, as for every effective version of Baker theorem, the constants ''d'' and 1/''c'' are so huge that this effective result can not be used in practice.
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| == Upper bounds for Diophantine approximations ==
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| ===General upper bound ===
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| {{main | Hurwitz's theorem (number theory)}}
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| The first important result about upper bounds for Diophantine approximations is [[Dirichlet's approximation theorem]], which implies that, for every irrational number {{math|''α''}}, there are infinitely many fractions <math>\tfrac{p}{q}\;</math> such that
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| : <math>\left|\alpha-\frac{p}{q}\right| < \frac{1}{q^2}\,.</math>
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| This implies immediately that one can not suppress the {{math|''ε''}} in the statement of Thue-Siegel-Roth theorem.
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| Over the years, this theorem has been improved until the following theorem of [[Émile Borel]] (1903).<ref>Perron (1913), Chapter 2, Theorem 15.</ref> For every irrational number {{math|''α''}}, there are infinitely many fractions <math>\tfrac{p}{q}\;</math> such that
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| : <math>\left|\alpha-\frac{p}{q}\right| < \frac{1}{\sqrt{5}q^2}\,.</math>
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| Therefore <math>\frac{1}{\sqrt{5}\, q^2}</math> is an upper bound for the Diophantine approximations of any irrational number.
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| The constant in this result may not be further improved without excluding some irrational numbers (see below).
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| === Equivalent real numbers ===
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| '''Definition''': Two real numbers <math>x,y</math> are called ''equivalent''<ref>A. Hurwitz: ''Ueber<!-- sic--> die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche''. In: ''Mathematische Annalen'', Band 39, 1891, S. 284.</ref><ref>G.H. Hardy & E.M. Wright, ''An Introduction to the Theory of Numbers'', 5th edition, 1979, Chapter 10.11.</ref> if there are integers <math>a,b,c,d\;</math> with <math>ad-bc = \pm 1\;</math> such that:
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| :<math>y = \frac{ax+b}{cx+d}\, .</math>
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| So equivalence is defined by an integer [[Möbius transformation]] on the real numbers, or by a member of the [[Modular group]] <math>\text{SL}_2^{\pm}(\Z)</math>, the set of invertible 2 × 2 matrices over the integers. Each rational number is equivalent to 0; thus the rational numbers are an [[equivalence class]] for this relation.
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| The equivalence may be read on the regular continued fraction representation, as shown by the following theorem of [[Serret]]:
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| '''Theorem''': Two irrational numbers ''x'' and ''y'' are equivalent if and only their exist two positive integers ''h'' and ''k'' such that the regular continued fraction representations of ''x'' and ''y''
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| :<math>x=[u_0; u_1, u_2, \ldots]\, ,</math>
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| :<math>y=[v_0; v_1, v_2, \ldots]\, ,</math>
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| verify
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| :<math>u_{h+i}=v_{k+i}</math>
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| for every non negative integer ''i''.<ref>See Perron (1929), Chapter 2, Theorem 23 (p. 63).</ref>
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| Thus, except for a finite initial sequence, equivalent numbers have the same continued fraction representation.
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| ===Lagrange spectrum ===
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| {{main|Markov spectrum}}
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| As said above, the constant in Borel's theorem may not improved, as shown by [[Adolf Hurwitz]] in 1891.<ref>[[G. H. Hardy]] and [[E. M. Wright]]: ''An Introduction to the Theory of Numbers'', 5th edition, Oxford University Press 1979, p 164</ref>
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| Let <math>\phi = \tfrac{1+\sqrt{5}}{2}</math> be the [[golden ratio]].
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| Then for any real constant ''c'' with <math>c > \sqrt{5}\;</math> there are only a finite number of rational numbers {{math|''p''/''q''}} such that
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| :<math>\left|\phi-\frac{p}{q}\right| < \frac{1}{c\, q^2}</math>.
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| Hence an improvement can only be achieved, if the numbers which are equivalent to <math>\phi</math> are excluded. More precisely:<ref>Cassels (1957) p.11</ref><ref>[[Adolf Hurwitz|A. Hurwitz]], 1891, Mathematische Annalen '''39''', ''Ueber<!-- sic--> die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche'', p. 279–284</ref>
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| For every irrational number <math>\alpha</math>, which is not equivalent to <math>\phi</math>, there are infinite many fractions <math>\tfrac{p}{q}\;</math> such that
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| : <math>\left|\alpha-\frac{p}{q}\right| < \frac{1}{\sqrt{8} q^2}.</math>
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| By successive exclusions — next one must exclude the numbers equivalent to <math>\sqrt 2</math> — of more and more classes of equivalence, the lower bound can be further enlarged.
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| The values which may be generated in this way are called ''Lagrange spectrum''.
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| They converge to the number 3 and are related to the [[Markov number]]s.<ref>Cassels (1957) p.18</ref><ref>See [http://www.math.jussieu.fr/~miw/articles/pdf/IntroductionDiophantineMethods.pdf Michel Waldschmidt: ''Introduction to Diophantine methods irrationality and transcendence''], pp 24–26.</ref>
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| == Khinchin's theorem and extensions == <!-- [[Khinchin's theorem on Diophantine approximations]] links here -->
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| [[Aleksandr Khinchin]] proved in 1926 that if <math>\phi</math>
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| is a non-increasing function from the positive integers to the positive real numbers such that <math>\sum_{q} \phi(q) < \infty\,, </math> then for almost all real numbers ''x'' (not necessarily algebraic), there are at most finitely many rational ''p''/''q'' and
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| :<math>\left| x- \frac{p}{q} \right| < \frac{\phi(q)}{|q|}.</math>
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| Similarly, if the sum diverges, then for almost all real numbers, there are infinitely many such rational numbers ''p''/''q''.<ref>Cassels (1957) p.120</ref><ref>Schmidt (1996) p.39</ref><ref>Khinchin (1997) pp.69–71</ref>
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| In 1941, R.J. Duffin and A.C. Schaeffer <ref>{{cite journal | zbl=0025.11002 | last1=Duffin | first1=R.J. | last2=Schaeffer | first2=A.C. | title=Khintchine's problem in metric diophantine approximation | journal=[[Duke Mathematical Journal]] | volume=8 | pages=243–255 | year=1941 | issn=0012-7094 }}</ref> proved a more general theorem that implies Khinchin's result, and made a conjecture now known by their name as the [[Duffin–Schaeffer conjecture]]. In 2006, V. Beresnevich and S. Velani proved a Hausdorff measure analogue of the conjecture, published in the [[Annals of Mathematics]].<ref>{{cite journal | zbl=1148.11033 | last1=Beresnevich | first1=Victor | last2=Velani | first2=Sanju | title=A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures | journal=[[Annals of Mathematics]] | volume=164 | year=2006 | pages=971–992 }}</ref>
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| == Uniform distribution ==
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| Another topic that has seen a thorough development is the theory of [[equidistributed sequence|uniform distribution mod 1]]. Take a sequence ''a''<sub>1</sub>, ''a''<sub>2</sub>, ... of real numbers and consider their ''fractional parts''. That is, more abstractly, look at the sequence in [[Quotient group|R/Z]], which is a circle. For any interval ''I'' on the circle we look at the proportion of the sequence's elements that lie in it, up to some integer ''N'', and compare it to the proportion of the circumference occupied by ''I''. ''Uniform distribution'' means that in the limit, as ''N'' grows, the proportion of hits on the interval tends to the 'expected' value. [[Hermann Weyl]] proved a basic result showing that this was equivalent to bounds for exponential sums formed from the sequence. This showed that Diophantine approximation results were closely related to the general problem of cancellation in exponential sums, which occurs throughout [[analytic number theory]] in the bounding of error terms.
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| Related to uniform distribution is the topic of [[irregularities of distribution]], which is of a [[combinatorics|combinatorial]] nature.
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| == Unsolved problems ==
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| There are still simply-stated unsolved problems remaining in Diophantine approximation, for example the ''[[Littlewood conjecture]]''.
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| It is also unknown if there are algebraic numbers with unbounded coefficients in their continued fraction expansion.
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| == Recent developments ==
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| In his plenary address at the [[International Mathematical Congress]] in Kyoto (1990), [[Grigory Margulis]] outlined a broad program rooted in [[ergodic theory]] that allows one to prove number-theoretic results using the dynamical and ergodic properties of actions of subgroups of [[semisimple Lie group]]s. The work of D.Kleinbock, G.Margulis, and their collaborators demonstrated the power of this novel approach to classical problems in Diophantine approximation. Among its notable successes are the proof of the decades-old [[Oppenheim conjecture]] by Margulis, with later extensions by Dani and Margulis and Eskin–Margulis–Mozes, and the proof of Baker and Sprindzhuk conjectures in the Diophantine approximations on manifolds by Kleinbock and Margulis. Various generalizations of the above results of [[Aleksandr Khinchin]] in metric Diophantine approximation have also been obtained within this framework.
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| ==See also==
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| * [[Davenport–Schmidt theorem]]
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| * [[Duffin–Schaeffer conjecture]]
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| * [[Low-discrepancy sequence]]
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| ==Notes==
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| {{Reflist|3}}
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| ==References==
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| * {{cite book | last=Bugeaud | first=Yann | title=Distribution modulo one and Diophantine approximation | series=Cambridge Tracts in Mathematics | volume=193 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2012 | isbn=978-0-521-11169-0 | zbl=pre06066616 }}
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| * {{cite book | author=J.W.S. Cassels | authorlink=J. W. S. Cassels | title=An introduction to Diophantine approximation | series=Cambridge Tracts in Mathematics and Mathematical Physics | volume=45 | publisher=[[Cambridge University Press]] | year=1957 }}
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| * {{cite book | first=A. Ya. | last=Khinchin | authorlink=Aleksandr Khinchin | title=Continued Fractions | publisher=Dover | year=1997 | origyear=1964 | isbn=0-486-69630-8 }}
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| *{{cite journal
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| | last1= Kleinbock | first1=D.Y.
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| | last2= Margulis | first2=G.A.
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| | author2-link = Grigory Margulis
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| | title = Flows on homogeneous spaces and Diophantine approximation on manifolds
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| | journal = Ann. Math.
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| | volume = 148
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| | issue = 1
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| | year = 1998
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| | pages = 339–360
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| | mr = 1652916 | zbl=0922.11061
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| | doi = 10.2307/120997
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| | jstor = 120997
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| }}
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| * {{cite book | first=Serge | last=Lang | authorlink=Serge Lang | title=Introduction to Diophantine Approximations | edition=New expanded | publisher=[[Springer-Verlag]] | year=1995 | isbn=0-387-94456-7 | zbl=0826.11030 }}
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| * {{cite book | first=G.A. | last=Margulis | authorlink=Grigory Margulis | chapter=Diophantine approximation, lattices and flows on homogeneous spaces | title=A panorama of number theory or the view from Baker's garden (Zürich, 1999) | editor1-last=Wüstholz | editor1-first=Gisbert | editor1-link=Gisbert Wüstholz | pages=280–310 | publisher=[[Cambridge University Press]] | location=Cambridge | year=2002 | mr=1975458 | zbl= | isbn=0-521-80799-9 }}
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| * {{cite book | zbl=0421.10019 | last=Schmidt | first=Wolfgang M. | authorlink=Wolfgang M. Schmidt | edition=1996 | title=Diophantine approximation | series=Lecture Notes in Mathematics | volume=785 | location=Berlin-Heidelberg-New York | publisher=[[Springer-Verlag]] | year=1980 | isbn=3-540-09762-7 }}
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| * {{cite book | last=Schmidt | first=Wolfgang M. | authorlink=Wolfgang M. Schmidt | title=Diophantine approximations and Diophantine equations | series=Lecture Notes in Mathematics | volume=1467 | publisher=[[Springer-Verlag]] | year=1996 | edition=2nd | isbn=3-540-54058-X | zbl=0754.11020 }}
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| * {{cite book | last=Sprindzhuk | first=Vladimir G. | title=Metric theory of Diophantine approximations | others=Transl. from the Russian and ed. by Richard A. Silverman. With a foreword by Donald J. Newman | series=Scripta Series in Mathematics | publisher=John Wiley & Sons | year=1979 | isbn=0-470-26706-2 | mr = 0548467 | zbl=0482.10047 }}
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| == External links ==
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| * [http://people.math.jussieu.fr/~miw/articles/pdf/HCMUNS10.pdf Diophantine Approximation: historical survey]. From ''Introduction to Diophantine methods'' course by [[Michel Waldschmidt]].
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| * {{springer|title=Diophantine approximations|id=p/d032600}}
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| {{DEFAULTSORT:Diophantine Approximation}}
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| [[Category:Number theory]]
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| [[Category:Diophantine approximation|*]]
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