发表机构
Faculty of Mechanics and Mathematics, Lomonosov Moscow State University（莫斯科国立大学力学与数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究一维Dirichlet谱中射线起点μ的上下界改进，证明Ivanov点Z_k为孤立点，并刻画其邻域间隙渐近行为及等价类。
AI 中文摘要
我们研究一维 Dirichlet 谱，并获得了关于谱中所含射线的起点 $\mu$ 及其孤立点的新结果。首先，我们证明了改进的上界 $\mu \leq (448875+1611\sqrt{67773})/46982 = 18.4809\ldots$。我们还展示了 Dirichlet 谱中的一个显式间隙 $(L,R)$，这给出了改进的下界 $\mu \geq R = 5.9596\ldots$。下界的证明包括对可容许连分数词进行精确的计算机辅助有限搜索。我们进一步证明了 Ivanov 引入的所有点 $Z_k$（$k\geq 2$）都是 Dirichlet 谱的孤立点。最后，我们确定了 $Z_k$ 周围相邻间隙的渐近行为，并描述了所有 Dirichlet 值为 $Z_k$ 的无理数的等价类。
英文摘要
We study the one-dimensional Dirichlet spectrum and obtain new results concerning the origin $μ$ of the ray contained in the spectrum and its isolated points. First, we prove the improved upper bound $μ\leq (448875+1611\sqrt{67773})/46982 = 18.4809\ldots$. We also exhibit an explicit gap $(L,R)$ in the Dirichlet spectrum, which yields the improved lower bound $μ\geq R = 5.9596\ldots$. The proof of the lower bound includes an exact computer-assisted finite search over admissible continued-fraction words. We further prove that all points $Z_k$, $k\geq 2$, introduced by Ivanov are isolated points of the Dirichlet spectrum. Finally, we determine the asymptotic behavior of the neighboring gaps around $Z_k$ and describe the equivalence classes of all irrational numbers whose Dirichlet value is $Z_k$.
