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Colóquio de Alunos

Metric diophantine approximation
Harold Erazo

Sexta-feira, 8 de novembro de 2019, 15:30
Auditorio 1

We know that any real number can be approximated by rational numbers with any degree of accuracy (just by truncating its decimal expansion), but how "quick" can we approximate a given $x \in\mathbb{R}$? Dirichlet's theorem says that every real number can be approximated at least with an order of $q^{-2}$ or in other words, that the inequality $\left| x - p/ q \right|<q^{-2}$ has infinitely many integer solutions $(p,q)$. How much can we improve this rate of approximation, say by $q^{-3}$? It turns out that  these numbers have Lebesgue measure 0, but they have Hausdorff dimension 2/3 (so they are not so small after all). 

The aim of the theory of metric diophantine approximation is to characterize number theoretic properties of typical numbers, by determining the size of exceptional sets. 

The objective of this talk is to review the basic results in this area, such as Khintchine's theorem, Jarnik's theorem, the Duffin-Schaeffer conjecture, etc and to sketch a proof of some results in order to give a flavor of the common techniques used here, as well as showing more recent developments like the Mass transference principle and finally, to explain how they fit in the more general framework of ubiquitous systems.