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Diophantine approximations and [[transcendental number theory]] are very close areas that share many theorems and methods. Diophantine approximations also have important applications in the study of [[Diophantine equation]]s.
The 2022 [[Fields Medal]] was awarded to [[James Maynard (mathematician)|James Maynard]], in part for his work on Diophantine approximation.
== Best Diophantine approximations of a real number ==
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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
== Lower bounds for Diophantine approximations ==
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== Uniform distribution ==
{{unsourced section|date=May 2023}}
Another topic that has seen a thorough development is the theory of [[equidistributed sequence|uniform distribution mod 1]]. Take a sequence ''a''<sub>1</sub>, ''a''<sub>2</sub>, ... of real numbers and consider their ''fractional parts''. That is, more abstractly, look at the sequence in <math>\mathbb{R}/\mathbb{Z}</math>, which is a circle. For any interval ''I'' on the circle we look at the proportion of the sequence's elements that lie in it, up to some integer ''N'', and compare it to the proportion of the circumference occupied by ''I''. ''Uniform distribution'' means that in the limit, as ''N'' grows, the proportion of hits on the interval tends to the 'expected' value. [[Hermann Weyl]] proved a [[Equidistributed_sequence#Weyl'
Related to uniform distribution is the topic of [[irregularities of distribution]], which is of a [[combinatorics|combinatorial]] nature.
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