An Improved Upper Bound for the Dirichlet Spectrum in Diophantine Approximation

TL;DR

This paper refines the upper bound for the Dirichlet spectrum's ray-origin constant by analyzing a Cantor-type set and applying sum-set results, resulting in a new bound of approximately 0.94866.

Contribution

It introduces a Cantor-type set with specific restrictions, analyzes its thickness, and uses sum-set results to improve the upper bound for the Dirichlet spectrum.

Findings

Established a new upper bound for the Dirichlet spectrum constant: approximately 0.94866.

Proved the thickness of a specific Cantor set exceeds 1.

Showed that the product of the Cantor set with itself forms an interval.

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 $\delta$. 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 $\tau(\log(F_4^*))>1$, 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 $\delta\le \frac{111(397+\sqrt{26565})}{65522}\approx0.94866$.