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arXiv · 1903.02882

Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum

Byungchul Cha·Dong Han Kim
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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.

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Byungchul Cha, Dong Han Kim. 2021-07-01. Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum. https://arxiv.org/abs/1903.02882

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