Editing Diophantine approximation (section)

== Khinchin's theorem on metric Diophantine approximation and extensions == <!-- [[Khinchin's theorem on Diophantine approximations]] links here -->

Let <math>\psi</math> be a positive real-valued function on positive integers  (i.e., a positive sequence) such that <math>q \psi(q)</math> is non-increasing. A real number ''x'' (not necessarily algebraic) is called <math>\psi</math>-''approximable'' if there exist infinitely many rational numbers  ''p''/''q'' such that

:<math>\left| x- \frac{p}{q} \right| < \frac{\psi(q)}{|q|}.</math>

[[Aleksandr Khinchin]] proved in 1926 that if the series <math display="inline">\sum_{q} \psi(q) </math> diverges, then almost every real number (in the sense of [[Lebesgue measure]]) is <math>\psi</math>-approximable, and if the series converges, then almost every real number is not <math>\psi</math>-approximable.  The circle of ideas surrounding this theorem and its relatives is known as ''metric Diophantine approximation'' or the ''metric theory of Diophantine approximation'' (not to be confused with height "metrics" in [[Diophantine geometry]]) or ''metric number theory''.

{{harvtxt|Duffin|Schaeffer|1941}} proved a generalization of Khinchin's result, and posed what is now known as the [[Duffin–Schaeffer conjecture]] on the analogue of  Khinchin's dichotomy for general, not necessarily decreasing, sequences <math>\psi</math> . {{harvtxt|Beresnevich|Velani|2006}} proved that a [[Hausdorff measure]] analogue of the Duffin–Schaeffer conjecture is equivalent to the original Duffin–Schaeffer conjecture, which is a priori weaker.
In July 2019, [[Dimitris Koukoulopoulos]] and [[James Maynard (mathematician)|James Maynard]] announced a proof of the conjecture.<ref>{{cite arXiv |first1=D. |last1=Koukoulopoulos |first2=J. |last2=Maynard |title=On the Duffin–Schaeffer conjecture |year=2019 |class=math.NT |eprint=1907.04593 }}</ref><ref>{{cite journal |last=Sloman |first=Leila |year=2019 |title=New Proof Solves 80-Year-Old Irrational Number Problem |journal=[[Scientific American]] |url=https://www.scientificamerican.com/article/new-proof-solves-80-year-old-irrational-number-problem/ }}</ref>

=== Hausdorff dimension of exceptional sets ===

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_\varepsilon(q) = \varepsilon q^{-1}</math>, where <math>\varepsilon > 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_\varepsilon</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.