Diophantine Approximation on a Circle

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Mathematics (CMC)

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The original problem of Diophantine approximation, which goes back to the famous 1842 theorem of Dirichlet, is that of approximating real numbers by rationals with controlled denominators. Since that time, various generalizations and extensions of Dirichlet's theorem have been proved in a number of different directions. In particular, these include Diophantine approximation with restricted fractions and simultaneous approximations for points in Euclidean spaces. We will start with the review of the classical Dirichlet-type results, and then move into Diophantine approximations by quotients of Pythagorean triples and make a connection to rational approximations for points on circles and ellipses. This talk will be completely accessible to undergraduate students.


This lecture was given during the Analysis Seminar at the Claremont Colleges in April 2009.

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© 2009 Lenny Fukshansky

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