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 $\hat{I}´$. Building on and refining V. A. Ivanov's approach, we introduce a Cantor-type set $F_4^{\ast }$ defined by certain restrictions on partial quotients. For its thickness, we prove $\ddot{I}„(\log (F_4^{\ast }))>1$, and apply sum-set results for Cantor sets to prove that the set $F_4^{\ast }\cdot F_4^{\ast }$ is an interval. Finally, we establish a new upper bound $\hat{I}´\le \frac{111(397+\sqrt{26565})}{65522}\approx 0.94866$.

16 pages