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arXiv · 2309.14646

Concentration of dimension in extremal points of left-half lines in the Lagrange spectrum

Carlos Gustavo Moreira·Christian Camilo Silva Villamil
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Abstract

We prove that for any $\eta$ that belongs to the closure of the interior of the Markov and Lagrange spectra, the sets $k^{-1}((-\infty,\eta])$ and $k^{-1}(\eta)$, which are the sets of irrational numbers with best constant of Diophantine approximation bounded by $\eta$ and exactly $\eta$ respectively, have the same Hausdorff dimension. We also show that, as $\eta$ varies in the interior of the spectra, this Hausdorff dimension is a strictly increasing function.

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Carlos Gustavo Moreira, Christian Camilo Silva Villamil. 2023-09-26. Concentration of dimension in extremal points of left-half lines in the Lagrange spectrum. https://arxiv.org/abs/2309.14646

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