Intrinsic Diophantine Approximation of circles

Abstract: Let $S^1$ be the unit circle in $\mathbb{R}^2$ centered at the origin and let $Z$ be a countable dense subset of $S^1$, for instance, the set $Z = S^1(\mathbb{Q})$ of all rational points in $S^1$. We give a complete description of an initial discrete part of the Lagrange spectrum of $S^1$ in the sense of intrinsic Diophantine approximation. This is an analogue of the classical result of Markoff in 1879, where he characterized the most badly approximable real numbers via the periods of their continued fraction expansions. Additionally, we present similar results for a few different subsets $Z$ of $S^1$. This is joint work with Dong Han Kim.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar