Gaps, isolated points, and the origin of the ray in the one-dimensional Dirichlet spectrum

Abstract

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 $Z_k$.