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Graduate Seminar

Zilin Jiang
Carnegie Mellon University
Title: Introduction to Diophantine Approximation

Abstract: Diophantine approximation deals with the approximation of real numbers by rational numbers. The problem to know the "best" approximation of a real number was solved during the 18th century by means of continued fractions. The main problem of the field is to find sharp lower bound of the difference between a real number and its rational approximation, expressed as a function of the denominator. More precisely, if $\alpha$ is an algebraic number of degree $n$ (at least 2) and the inequality $$\left|\alpha - \frac{h}{q}\right| < \frac{1}{q^\kappa}$$ is satisfied by infinitely many rational numbers $h/q$, then, historically, Thue in 1908 proved that $\kappa\leq \frac{1}{2}n+1$, Siegel in 1921 proved that $\kappa \leq s + \frac{n}{s+1}$ for $s=1,2,\ldots, n-1$, Dyson in 1947 proved that $\kappa\leq\sqrt{2n}$, and Roth in 1955 proved that $\kappa\leq 2$ for which he was awarded the Fields medal in 1958.

In this talk, I will start with Liouville's remark on rational approximation to algebraic numbers and its relation with transcendental theory, and then present Thue's work on Diophantine approximation. The exposition assumes minimal knowledge of number theory and aims to be accessible to the audience with standard mathematical background. Some applications in Diophantine equation and Baker's further development on the subject will be mentioned if time permits.

Date: Tuesday, November 4, 2014
Time: 5:30 pm
Location: Wean Hall 8220
Submitted by:  Zilin Jiang