The normalized approximation function for multiple zeta-star values

Technical Summary: The Normalized Approximation Function for Multiple Zeta-Star Values

Problem Statement
The paper addresses the quantitative Diophantine approximation problem arising from the zeta-star correspondence, a bijection $\eta$ between infinite admissible sequences and the real interval $(1, +\infty)$. While the set of finite multiple zeta-star values, $Z_\star$, is known to be dense in $(1, +\infty)$, unnormalized approximation is trivial. The central problem is to define a meaningful approximation metric that accounts for the complexity of the approximating indices. The author introduces a normalized approximation function $N(\alpha)$, analogous to the reciprocal Lagrange constant in continued fraction theory, to measure the asymptotic quality of approximation by $Z_\star$ relative to the intrinsic binary scale of the indices.

Methodology and Framework
The analysis relies on the interplay between the order structure of multiple zeta-star values, their binary coordinate representation, and metric number theory.

1. Zeta-Star Correspondence and Binary Coordinates:
   The paper utilizes the bijection $\eta: T \to (1, +\infty)$, where $T$ is the set of infinite admissible sequences. A key tool is the binary coordinate $\beta(k) = \sum 2^{-K_j}$ (where $K_j$ is the partial weight), which maps $T$ to $(0, 1/2)$. The function $\Phi = \eta \circ \beta^{-1}$ maps the binary interval to $(1, +\infty)$. The derivative of $\Phi$, denoted $\Phi'$, plays a crucial role in relating binary approximation errors to zeta-star approximation errors.

2. Definition of the Normalized Function:
   For a real number $\alpha > 1$ and a depth $r$, let $S_r$ be the set of admissible indices of depth $r$. The depthwise approximation error is defined as:
   $$\Delta_r(\alpha) := \inf_{k \in S_r} 2^{\text{wt}(k)} |\alpha - \zeta^\star(k)|$$
   where $\text{wt}(k)$ is the weight of the index. The normalized approximation function is the limit inferior:
   $$N(\alpha) := \liminf_{r \to \infty} \Delta_r(\alpha)$$
   This definition normalizes the error by the binary scale $2^{-\text{wt}(k)}$ associated with the symbolic cylinder of the index.

3. Analytical Tools:

   • Upper Semicontinuity: The paper establishes that $\Delta_r(\alpha)$ is upper semicontinuous, implying $N(\alpha)$ is Borel measurable.
   • Derivative Estimates: Using results from Hirose, Murahara, and Onozuka, the paper derives explicit formulas for $\Phi'(y)$ involving infinite sums of multiple zeta-star type terms.
   • Metric Theory: The analysis employs Borel–Cantelli lemmas and divergence conditions to determine the measure of sets where approximation is "good" (i.e., $N(\alpha)$ is small).

Key Contributions and Results

1. Topological and Regularity Properties

• Upper Semicontinuity: The map $\alpha \mapsto \Delta_r(\alpha)$ is upper semicontinuous for each fixed depth $r$.
• Measurability: The function $N(\alpha)$ is Borel measurable.
• Residual Zero Set: The set $\{\alpha > 1 : N(\alpha) = 0\}$ is a dense $G_\delta$ subset of $(1, +\infty)$. This implies that "generic" points in the Baire category sense are exceptionally well-approximated.

2. Metric Results (Almost Everywhere Behavior)

• Convergence: If $\sum 2^{-g(r)} < \infty$, the set of $\alpha$ where $\Delta_r(\alpha) < 2^{-g(r)}$ infinitely often has Lebesgue measure zero.
• Divergence: Under a logarithmically reinforced divergence condition ($\sum \frac{2^{-g(r)}}{r \log^2 r} = \infty$), the complement of the approximation set has measure zero.
• Main Metric Theorem: $N(\alpha) = 0$ for Lebesgue almost every $\alpha > 1$. This is proven by combining the divergence theorem with the existence of infinitely many large "next digits" in the zeta-star expansion for almost all $\alpha$.

3. Evaluations at Specific Points

• Finite Multiple Zeta-Star Values: For a finite admissible index $k$, the value $N(\zeta^\star(k))$ is explicitly calculated as the normalized length of the associated basic interval, $L(k)$.
  • If $k = (2, \{1\}^{s-1})$, then $N(\zeta^\star(k)) = +\infty$.
  • Otherwise, $N(\zeta^\star(k))$ is finite and strictly greater than 1.
• Integers: For every integer $n \ge 2$, $N(n) = n$. This is derived using the arithmetic properties of the binary expansion of $n$ and the derivative of $\Phi$.
• Discontinuity: The function $\alpha \mapsto N(\alpha)$ is nowhere continuous on $(1, +\infty)$ when the codomain is equipped with the order topology. This is due to the coexistence of points where $N(\alpha)=0$ (dense) and points where $N(\alpha) > 1$ or $+\infty$ (dense).

4. Image of the Function

• The image of $N$, denoted $\text{Im}(N)$, is dense in $[0, +\infty]$.
• Specifically, the set of finite values $\{N(\alpha) : N(\alpha) < \infty\}$ is dense in $[0, +\infty)$. The paper constructs a family of values $S_q / (2m-1)$ that approach any target $t \ge 0$.

Significance and Claims
The paper claims to establish the regularity and generic behavior of the normalized approximation function for multiple zeta-star values. By drawing an analogy to the classical Lagrange spectrum in continued fractions, the work:

1. Defines a natural scale for Diophantine approximation in the context of multiple zeta-star values.
2. Proves that, unlike the classical Lagrange spectrum which has a discrete lower part, the "spectrum" of $N(\alpha)$ is dominated by the value 0 almost everywhere, yet the function takes a wide range of values on a dense set.
3. Provides a complete characterization of $N(\alpha)$ at specific points (integers and finite zeta-star values), revealing a stark contrast between the generic behavior (0) and the behavior at "rational-like" points (finite zeta-star values and integers).

The paper concludes by proposing conjectures regarding the structure of the approximation spectrum $L_\star = \{1/N(\alpha) : 0 < N(\alpha) < \infty\}$, suggesting it contains a half-line $(c, +\infty)$ and has a non-empty complement, mirroring the complex structure of the classical Lagrange spectrum. The work leaves open the determination of the exact image set and the Hausdorff dimensions of the level sets.