Diophantine approximation: Difference between revisions

Revision as of 09:35, 5 January 2025

Revision as of 11:27, 10 January 2025

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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 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]\, , \\