On the transference principle in Diophantine approximation

Abstract: In 1842, Dirichlet published his famous theorem which became the foundation of Diophantine approximation. The phenomenon he found inspired Liouville to study how well algebraic numbers can be approximated by rationals, and thus, to come up with a method of constructing transcendental numbers explicitly. The development of these ideas led to the concepts of irrationality measure and transcendence measure. Thanks to Minkowski, it became clear that many problems arising in the theory of Diophantine approximation could be addressed quite effectively using the tools of geometry of numbers. In particular, the geometric approach naturally offers a wide variety of multidimensional analogues of the concept of irrationality measure — so called Diophantine exponents. In the talk, we will discuss various Diophantine exponents and the geometry that arises when studying them. We will pay special attention to the phenomenon discovered by Khintchine, which he called the transference principle.

algebraic geometrynumber theory

Audience: researchers in the topic

FGC-HRI-IPM Number Theory Webinars

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