# Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum

## TL;DR

This paper characterizes the structure of the Lagrange spectrum for intrinsic Diophantine approximation on the unit circle, identifying the smallest accumulation point and relating points with low Lagrange numbers to Romik's digit expansions.

## Contribution

It provides a complete description of the Lagrange spectrum on the circle below its smallest accumulation point, extending Markoff's theorem to this geometric setting.

## Key findings

- Smallest accumulation point of the spectrum is 2.
- Points with Lagrange numbers less than 2 are characterized by Romik's digit expansions.
- The structure of the spectrum below the smallest accumulation point is fully described.

## Abstract

Let $\mathscr{L}(S^1)$ be the Lagrange spectrum arising from intrinsic Diophantine approximation on the unit circle $S^1$ by its rational points. We give a complete description of the structure of $\mathscr{L}(S^1)$ below its smallest accumulation point. To this end, we use digit expansions of points on $S^1$, which were originally introduced by Romik in 2008 as an analogue of simple continued fraction of a real number. We prove that the smallest accumulation point of $\mathscr{L}(S^1)$ is 2. Also we characterize the points on $S^1$ whose Lagrange numbers are less than 2 in terms of Romik's digit expansions. Our theorem is the analogue of the celebrated theorem of Markoff on badly approximable real numbers.