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

TL;DR

This paper investigates the Hausdorff dimension of sets of irrationals with bounded Diophantine approximation constants within the Lagrange spectrum, revealing their equal dimensions at certain points and a strictly increasing dimension function.

Contribution

It establishes that these sets have equal Hausdorff dimensions at boundary points and shows the dimension varies strictly monotonically within the spectrum.

Findings

Sets at boundary points have equal Hausdorff dimension.

Hausdorff dimension is strictly increasing as ta varies.

Provides new insights into the structure of the Lagrange spectrum.

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.