AI 中文总结
该研究受拉格朗日谱及其连分数倒数形式启发，引入多重zeta星值的归一化逼近函数，证明其正则性、可测性等性质，得出几乎所有大于1的实数对应函数值为0且像集稠密的结论。
AI 中文摘要
受经典拉格朗日谱及连分数理论中拉格朗日谱倒数形式的启发，我们引入了用于多重zeta星值逼近的归一化逼近函数$\boldsymbol{\textit{N}}(\boldsymbol{\textit{\u03b1}})$。在每个深度层级，我们对所有容许指标的逼近误差进行最小化，并根据其权重确定的二进制尺度进行归一化。随后，通过取深度趋于无穷时的下极限，得到函数$\boldsymbol{\textit{N}}(\boldsymbol{\textit{\u03b1}})$。利用多重zeta星值的序结构，我们建立了该函数的正则性与一般行为：证明了深度级逼近函数是上半连续的，$\boldsymbol{\textit{N}}$是Borel可测的，且其零集是区间$(1,+\boldsymbol{\u221e})$中的稠密$G_\boldsymbol{\u03b4}$子集。我们还推导了一个自然前缀逼近估计，表明较大的下一位数字会产生异常良好的归一化逼近。结合作者关于多重zeta星值丢番图逼近的度量结果，这给出了对数强化发散条件下的全测度定理，特别地，证明了对Lebesgue测度下几乎所有的$\boldsymbol{\u03b1}>1$，都有$\boldsymbol{\textit{N}}(\boldsymbol{\textit{\u03b1}})=0$。最后，我们给出了归一化逼近函数像的一些基本性质，并证明其像$\boldsymbol{\text{Im}}(\boldsymbol{\textit{N}})$在扩展半直线$[0,+\boldsymbol{\u221e}]$上是稠密的。
英文摘要
Motivated by the classical Lagrange spectrum and the reciprocal formulation of the Lagrange spectrum in continued-fraction theory, we introduce a normalized approximation function $\mathcal{N}(α)$ for approximation by multiple zeta-star values. At each depth, the approximation error is minimized over all admissible indices and normalized by the binary scale determined by their weights. The function $\mathcal{N}(α)$ is then obtained by taking the limit inferior as the depth tends to infinity. Using the order structure of multiple zeta-star values, we establish the regularity and generic behavior of this function. We prove that the depthwise approximation functions are upper semicontinuous, that $\mathcal{N}$ is Borel measurable, and that its zero set is a dense $G_δ$ subset of $(1,+\infty)$. We also derive a natural-prefix approximation estimate showing that a large next digit produces an exceptionally good normalized approximation. Combined with the author's metric resultson Diophantine approximation of multiple zeta-star values, this gives a full-measure theorem under a logarithmically reinforced divergence condition and, in particular, proves that $\mathcal{N}(α)=0$ for Lebesgue almost every $α>1$. Finally, we give some basic properties of the image of the normalized approximation function and show that the image $\mathrm{Im}(\mathcal{N})$ is dense on the extended half-line $[0,+\infty]$.
Comments31 pages
