Lagrange谱中间隙的允许端点
Admissible endpoints of gaps in the Lagrange spectrum
摘要：
我们称一个正实数$λ$是可容许的，如果它属于Lagrange谱，并且存在一个无理数$α$使得$μ(α)=λ$ .这里$μ(α)$表示$α$的拉格朗日常数-最大实数$c$使得$\forall \varepsilon>0$不等式$|α-\frac{p}{q}|\le\frac{1}{(c-\varepsilon)q^2}$对于互质的$p$和$q$有无穷多个解。本文建立了Lagrange谱元可容许的一个充要条件，并构造了一个不可容许数的无穷级数。
英文摘要：
We call a positive real number $λ$ admissible if it belongs to the Lagrange spectrum and there exists an irrational number $α$ such that $μ(α)=λ$. Here $μ(α)$ denotes the Lagrange constant of $α$ - maximal real number $c$ such that $\forall \varepsilon>0$ the inequality $|α-\frac{p}{q}|\le\frac{1}{(c-\varepsilon)q^2}$ has infinitely many solutions for relatively prime $p$ and $q$. In this paper we establish a necessary and sufficient condition of admissibility of the Lagrange spectrum element and construct an infinite series of not admissible numbers.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
