Diophantine approximation: Difference between revisions

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In [[number theory]], the study of '''Diophantine approximation''' deals with the approximation of [[real number]]s by [[rational number]]s. It is named after [[Diophantus of Alexandria]].

The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number ''ap''/''bq'' is a "good" approximation of a real number ''α'' if the absolute value of the difference between ''ap''/''bq'' and ''α'' may not decrease if ''ap''/''bq'' is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of [[simple continued fraction]]s.

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]]. 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]].

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Diophantine approximations and [[transcendental number theory]] are very close areas that share many theorems and methods. Diophantine approximations also have important applications in the study of [[Diophantine equation]]s.

The 2022 [[Fields Medal]] was awarded to [[James Maynard (mathematician)|James Maynard]], in part for his work on Diophantine approximation.

== Best Diophantine approximations of a real number ==

{{main|ContinuedSimple continued fraction#Best rational approximations|Irrationality measure}}

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">{{harvnb|Khinchin|1997|p=21}}</ref> the rational number {{math|''p''/''q''}} is a ''best Diophantine approximation'' of {{math|''α''}} if

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A best approximation for the second definition is also a best approximation for the first one, but the converse is not true in general.<ref name=Khinchin24>{{harvnb|Khinchin|1997|p=24}}</ref>

The theory of [[Simple continued fraction|continued fraction]]s allows us to compute the best approximations of a real number: for the second definition, they are the [[convergentSimple (continued fraction)#Convergents|convergents]] of its expression as a regular continued fraction.<ref name=Lang9/><ref name=Khinchin24/><ref>{{harvnb|Cassels|1957|pp=5–8}}</ref> For the first definition, one has to consider also the [[ContinuedSimple continued fraction#Semiconvergents|semiconvergents]].<ref name="Khinchin 1997 p.21"/>

For example, the constant ''e'' = 2.718281828459045235... has the (regular) continued fraction representation

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The badly approximable numbers are precisely those with [[Restricted partial quotients|bounded partial quotients]].<ref name=Bug245>{{harvnb|Bugeaud|2012|p=245}}</ref>

Equivalently, a number is badly approximable [[if and only if]] its [[Markov constant]] is finite or equivalently its simple continued fraction is bounded.

== Lower bounds for Diophantine approximations ==

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{{main|Thue–Siegel–Roth theorem}}

Over more than a century, there were many efforts to improve Liouville's theorem: every improvement of the bound enables us to prove that more numbers are transcendental. The main improvements are due to {{harvs|first=Axel|last=Thue|authorlink=Axel Thue|year=1909|txt}}, {{harvs|frst=Carl Ludwig|last=Siegel|authorlink=Carl Ludwig Siegel|year=1921|txt}}, {{harvs|first=Freeman|last=Dyson|authorlink=Freeman Dyson|year=1947|txt}}, and {{harvs|first=Klaus|last=Roth|authorlink=Klaus Roth|year=1955|txt}}, leading finally to the Thue–Siegel–Roth theorem: If {{math|''x''}} is an irrational algebraic number and {{math|''ε > 0''}} a (small) positive real number, then there exists a positive constantreal number {{math|''c''(''x'', ''ε'')}} such that

:<math>

\left| x- \frac{p}{q} \right|>\frac{c(x, \varepsilon)}{q^{2+\varepsilon}}

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The equivalence may be read on the regular continued fraction representation, as shown by the following theorem of [[Joseph Alfred Serret|Serret]]:

'''Theorem''': Two irrational numbers ''x'' and ''y'' are equivalent if and only if there exist two positive integers ''h'' and ''k'' such that the regular [[Simple continued fraction|continued fraction]] representations of ''x'' and ''y''

:<math>\begin{align}

x &= [u_0; u_1, u_2, \ldots]\, , \\

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== Uniform distribution ==

{{unsourced section|date=May 2023}}

Another topic that has seen a thorough development is the theory of [[equidistributed sequence|uniform distribution mod 1]]. Take a sequence ''a''₁, ''a''₂, ... of real numbers and consider their ''fractional parts''. That is, more abstractly, look at the sequence in <math>\mathbb{R}/\mathbb{Z}</math>, 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 [[Equidistributed_sequence#Weyl's criterions_criterion|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.

Related to uniform distribution is the topic of [[irregularities of distribution]], which is of a [[combinatorics|combinatorial]] nature.

== Algorithms ==

Grotschel, Lovasz and Schrijver describe algorithms for finding approximately-best diophantine approximations, both for individual real numbers and for set of real numbers. The latter problem is called '''simultaneous diophantine approximation'''.<ref name=":0">{{Cite Geometric Algorithms and Combinatorial Optimization}}</ref>{{Rp|page=|___location=Sec. 5.2}}

== Unsolved problems ==

{{unsourced section|date=May 2023}}

There are still simply- stated unsolved problems remaining in Diophantine approximation, for example the ''[[Littlewood conjecture]]'' and the ''[[lonely runner conjecture]]''.

It is also unknown if there are algebraic numbers with unbounded coefficients in their continued fraction expansion.

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* [[Davenport–Schmidt theorem]]

* [[Duffin–Schaeffer conjecturetheorem]]

* [[Heilbronn set]]

* [[Low-discrepancy sequence]]

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| volume = 42

| year = 2013

| isbn = 978-3-642-36067-1

| s2cid = 55652124

}}

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== External links ==

* [http://people.math.jussieu.fr/~miw/articles/pdf/HCMUNS10.pdf Diophantine Approximation: historical survey] {{Webarchive|url=https://web.archive.org/web/20120214101838/http://people.math.jussieu.fr/~miw/articles/pdf/HCMUNS10.pdf |date=2012-02-14 }}. From ''Introduction to Diophantine methods'' course by [[Michel Waldschmidt]].

* {{springer|title=Diophantine approximations|id=p/d032600}}