Diophantine approximation: Difference between revisions

Revision as of 03:38, 16 November 2022

Revision as of 14:19, 15 May 2023

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{{Short description|Rational-number approximation of a real number}}

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{{Use American English|date = March 2019}}

{{Diophantine_approximation_graph.svg}}

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== Lower bounds for Diophantine approximations ==

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=== Approximation of a rational by other rationals ===

A rational number <math display="inline">\alpha =\frac{a}{b}</math> may be obviously and perfectly approximated by <math display="inline">\frac{p_i}{q_i} = \frac{i\,a}{i \,b}</math> for every positive integer ''i''.

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== Uniform distribution ==

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Another topic that has seen a thorough development is the theory of [[equidistributed sequence|uniform distribution mod 1]]. Take a sequence ''a''₁, ''a''₂, ... 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 [[Weyl's criterion|basic result]] showing that this was equivalent to bounds for exponential sums formed from the sequence. This showed that Diophantine approximation results were closely related to the general problem of cancellation in exponential sums, which occurs throughout [[analytic number theory]] in the bounding of error terms.

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== Unsolved problems ==

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There are still simply-stated unsolved problems remaining in Diophantine approximation, for example the ''[[Littlewood conjecture]]'' and the ''[[lonely runner conjecture]]''.

It is also unknown if there are algebraic numbers with unbounded coefficients in their continued fraction expansion.

== Recent developments ==

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In his plenary address at the [[International Mathematical Congress]] in Kyoto (1990), [[Grigory Margulis]] outlined a broad program rooted in [[ergodic theory]] that allows one to prove number-theoretic results using the dynamical and ergodic properties of actions of subgroups of [[semisimple Lie group]]s. The work of D. Kleinbock, G. Margulis and their collaborators demonstrated the power of this novel approach to classical problems in Diophantine approximation. Among its notable successes are the proof of the decades-old [[Oppenheim conjecture]] by Margulis, with later extensions by Dani and Margulis and Eskin–Margulis–Mozes, and the proof of Baker and Sprindzhuk conjectures in the Diophantine approximations on manifolds by Kleinbock and Margulis. Various generalizations of the above results of [[Aleksandr Khinchin]] in metric Diophantine approximation have also been obtained within this framework.