Editing Diophantine approximation (section)

== Upper bounds for Diophantine approximations ==

===General upper bound ===
{{main | Hurwitz's theorem (number theory)}}

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
: <math>\left|\alpha-\frac{p}{q}\right| < \frac{1}{q^2}\,.</math>

This implies immediately that one cannot suppress the {{math|''ε''}} in the statement of Thue-Siegel-Roth theorem.

[[Adolf Hurwitz]] (1891)<ref>{{harvnb|Hurwitz|1891|p=279}}</ref> strengthened this result, proving that for every irrational number {{math|''α''}}, there are infinitely many fractions <math>\tfrac{p}{q}\;</math> such that
: <math>\left|\alpha-\frac{p}{q}\right| < \frac{1}{\sqrt{5}q^2}\,.</math>

Therefore, <math>\frac{1}{\sqrt{5}\, q^2}</math> is an upper bound for the Diophantine approximations of any irrational number.
The constant in this result may not be further improved without excluding some irrational numbers (see below).

[[Émile Borel]] (1903)<ref>{{harvnb|Perron|1913|loc=Chapter 2, Theorem 15}}</ref> showed that, in fact, given any irrational number {{math|''α''}}, and given three consecutive convergents of {{math|''α''}}, at least one must satisfy the inequality given in Hurwitz's Theorem.

=== Equivalent real numbers ===

'''Definition''': Two real numbers <math>x,y</math> are called ''equivalent''<ref>{{harvnb|Hurwitz|1891|p=284}}</ref><ref>{{harvnb|Hardy|Wright|1979|loc=Chapter 10.11}}</ref> if there are integers <math>a,b,c,d\;</math> with <math>ad-bc = \pm 1\;</math> such that:
:<math>y = \frac{ax+b}{cx+d}\, .</math>

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 &times; 2 matrices over the integers. Each rational number is equivalent to 0; thus the rational numbers are an [[equivalence class]] for this relation.

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]\, , \\
  y &= [v_0; v_1, v_2, \ldots]\, ,
\end{align}</math>

satisfy
:<math>u_{h+i} = v_{k+i}</math>
for every non negative integer ''i''.<ref>See {{harvnb|Perron|1929|loc=Chapter 2, Theorem 23, p. 63}}</ref>

Thus, except for a finite initial sequence, equivalent numbers have the same continued fraction representation.

Equivalent numbers are approximable to the same degree, in the sense that they have the same [[Markov constant]].

===Lagrange spectrum ===
{{main|Markov spectrum}}
As said above, the constant in Borel's theorem may not be improved, as shown by [[Adolf Hurwitz]] in 1891.<ref>{{harvnb|Hardy|Wright|1979|p=164}}</ref>
Let <math>\phi = \tfrac{1+\sqrt{5}}{2}</math> be the [[golden ratio]].
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 
:<math>\left|\phi-\frac{p}{q}\right| < \frac{1}{c\, q^2}.</math>

Hence an improvement can only be achieved, if the numbers which are equivalent to <math>\phi</math> are excluded. More precisely:<ref>{{harvnb|Cassels|1957|p=11}}</ref><ref>{{harvnb|Hurwitz|1891}}</ref>
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

: <math>\left|\alpha-\frac{p}{q}\right| < \frac{1}{\sqrt{8} q^2}.</math>

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.
The values which may be generated in this way are ''Lagrange numbers'', which are part of the [[Markov spectrum|Lagrange spectrum]]. 
They converge to the number 3 and are related to the [[Markov number]]s.<ref>{{harvnb|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''] {{Webarchive|url=https://web.archive.org/web/20120209111526/http://www.math.jussieu.fr/~miw/articles/pdf/IntroductionDiophantineMethods.pdf |date=2012-02-09 }}, pp 24–26.</ref>