On discrete part of Dirichlet spectrum

TL;DR

This work transfers Hančl's improvements from the Lagrange spectrum to the Dirichlet spectrum, focusing on its discrete part. It establishes a refined lower envelope $f_0$ governing the limsup behavior of $t\cdot\psi_{\alpha}(t)$ near the minimal Dirichlet value, with an exact extremal characterization: equality in the refined bound occurs only for $\alpha=\pm\alpha_0+C$ (up to integers). Extending the analysis to the full discrete Dirichlet spectrum, the paper introduces a family of envelopes $f_k$ tied to the extremal numbers $\alpha_k$ and their pre-periodic variants $\beta_k$, $\beta_k^{(1)}$, $\beta_k^{(2)}$, and proves that relevant $\alpha$-values attain $\lim_{t\to q_n-} t\psi_{\alpha}(t)$ arbitrarily close to these envelopes via subsequences of convergents. The approach hinges on Perron-type formulas, precise continued-fraction inequalities, and a careful case analysis to identify when the refined bounds are sharp, thereby enriching the understanding of the discrete Dirichlet spectrum and its extremal elements.

Abstract

Recently J.Hančl obtained a result which improves on approximations to real numbers which correspond to the discrete part of Lagrange spectrum. In the present paper we prove a similar result related to the discrete part of Dirichlet spectrum.