The normalized approximation function for multiple zeta-star values

Abstract

Motivated by the classical Lagrange spectrum and the reciprocal formulation of the Lagrange spectrum in continued-fraction theory, we introduce a normalized approximation function $\mathcal{N}(\alpha)$ 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 $\mathcal{N}(\alpha)$ 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 $G_\delta$ 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 $\mathcal{N}(\alpha)=0$ for Lebesgue almost every $\alpha>1$. 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 $[0,+\infty]$.