Diophantine approximation: Difference between revisions

Revision as of 22:13, 25 September 2022

Revision as of 22:14, 25 September 2022

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An important example of a function <math>\psi</math> to which Khinchin's theorem can be applied is the function <math>\psi_c(q) = q^{-c}</math>, where ''c''&nbsp;>&nbsp;1 is a real number. For this function, the relevant series converges and so Khinchin's theorem tells us that almost every point is not <math>\psi_c</math>-approximable. Thus, the set of numbers which are <math>\psi_c</math>-approximable forms a subset of the real line of Lebesgue measure zero. The Jarník-Besicovitch theorem, due to [[Vojtech Jarnik|V. Jarník]] and [[Abram Samoilovitch Besicovitch|A. S. Besicovitch]], states that the [[Hausdorff dimension]] of this set is equal to <math>1/c</math>.<ref>{{harvnb|Bernik|Beresnevich|Götze|Kukso|2013|p=24}}</ref> In particular, the set of numbers which are <math>\psi_c</math>-approximable for some <math>c > 1</math> (known as the set of ''very well approximable numbers'') has Hausdorff dimension one, while the set of numbers which are <math>\psi_c</math>-approximable for all <math>c > 1</math> (known as the set of [[Liouville number]]s) has Hausdorff dimension zero.

Another important example is the function <math>\psi_\epsilonvarepsilon(q) = \epsilonvarepsilon q^{-1}</math>, where <math>\epsilonvarepsilon > 0</math> is a real number. For this function, the relevant series diverges and so Khinchin's theorem tells us that almost every number is <math>\psi_\epsilonvarepsilon</math>-approximable. This is the same as saying that every such number is ''well approximable'', where a number is called well approximable if it is not badly approximable. So an appropriate analogue of the Jarník-Besicovitch theorem should concern the Hausdorff dimension of the set of badly approximable numbers. And indeed, V. Jarník proved that the Hausdorff dimension of this set is equal to one. This result was improved by [[Wolfgang M. Schmidt|W. M. Schmidt]], who showed that the set of badly approximable numbers is ''incompressible'', meaning that if <math>f_1,f_2,\ldots</math> is a sequence of [[Lipschitz continuity#Lipschitz manifolds|bi-Lipschitz]] maps, then the set of numbers ''x'' for which <math>f_1(x),f_2(x),\ldots</math> are all badly approximable has Hausdorff dimension one. Schmidt also generalized Jarník's theorem to higher dimensions, a significant achievement because Jarník's argument is essentially one-dimensional, depending on the apparatus of continued fractions.

== Uniform distribution ==