Diophantine approximation is a branch of number theory where one studies approximation of irrational numbers by rationals and quality of such approximations. In this talk, we will consider intrinsic Diophantine approximation, which deals with approximating irrational points in a closed subset $X$ in $\mathbb{R}^n$ via rational points lying in $X$. First, we consider $X = S^1$, the unit circle in $\mathbb{R}^2$ centered at the origin. 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 approximations of $S^1$ by a few different sets of rational points. This is joint work with Dong Han Kim (Dongguk University, Seoul). (Contact Bo-Hae Im if you plan to join the seminar.)
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