An Improved Upper Bound for the Dirichlet Spectrum in Diophantine Approximation

Abstract

We study the continuous part of the Dirichlet spectrum $\mathbb{D}$ and improve the best previously published upper bound for the ray-origin constant $δ$. Building on and refining V. A. Ivanov's approach, we introduce a Cantor-type set $F_4^*$ defined by certain restrictions on partial quotients. For its thickness, we prove $τ(\log(F_4^*))>1$, and apply sum-set results for Cantor sets to prove that the set $F_4^* \cdot F_4^*$ is an interval. Finally, we establish a new upper bound $δ\le \frac{111(397+\sqrt{26565})}{65522}\approx0.94866$.