Editing Diophantine approximation (section)

==Measure of the accuracy of approximations ==

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 display="inline">\left|\alpha - \frac{p}{q}\right|.</math> However, this quantity can 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.

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 display="inline">\left|\alpha - \frac{p}{q}\right|>\phi(q)</math> ". In some cases, "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''}}.

For upper bounds, one has to take into account 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 display="inline">\left|\alpha - \frac{p}{q}\right|<\phi(q)</math> ".

===Badly approximable numbers===
A '''badly approximable number''' is an ''x'' for which there is a positive constant ''c'' such that for all rational ''p''/''q'' we have

:<math>\left|{ x - \frac{p}{q} }\right| > \frac{c}{q^2} \ . </math>

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.