Weighted error-sum identities for periodic continued fractions and their generalizations

Technical Summary: Weighted Error-Sum Identities for Periodic Continued Fractions and Their Generalizations

Problem Statement
The paper addresses the study of approximation errors arising from the convergents of continued fractions. For a real number $\xi$ with convergents $h_n/k_n$, the local approximation error is defined as $\varepsilon_n = h_n - \xi k_n$. While the asymptotic behavior of these errors is well-understood, the paper focuses on the global structure of weighted error sums, specifically the series:
$$f_\xi(s) = \sum_{n=-1}^{\infty} a_{n+1} |h_n - \xi k_n|^s$$
for $s \ge 1$. The primary problem is to derive explicit, closed-form expressions for these sums when $\xi$ is a purely periodic quadratic irrational (i.e., $\xi = [a_0, a_1, \dots, a_{N-1}]$). Additionally, the authors seek to extend these results to generalized continued fractions with non-unit numerators to derive identities for mathematical constants such as $\pi$ and $\ln 2$.

Methodology
The authors employ a combination of linear algebra, classical number theory, and the theory of continued fractions. The core of their methodology relies on the observation that for purely periodic continued fractions, the sequence of approximation errors decomposes into $N$ geometric subsequences.

The paper presents three independent proofs for this geometric decomposition (Theorem 2.1):

1. Matrix Dynamics: By defining cyclic period matrices $M_r$ acting on the vector of consecutive errors $(\varepsilon_n, \varepsilon_{n-1})^T$, the authors show that these matrices share eigenvalues. The contraction of the error vector is governed by the eigenvalue $\rho$, which is the conjugate of a specific algebraic unit $u$.
2. Algebraic Conjugation: Using the quadratic equation satisfied by $\xi$ and Vieta's formulas, the authors derive a recurrence relation for the errors directly, showing $\varepsilon_{n+N} = \rho \varepsilon_n$ without explicit matrix notation.
3. Complete Quotients: By analyzing the sequence of complete quotients $\xi_n$ and their product over a period, the authors relate the error terms to the product of these quotients, again establishing the geometric progression.

Once the geometric nature of the error subsequences is established ($\varepsilon_{mN+r} = \varepsilon_r \rho^m$), the weighted sums $f_\xi(s)$ are evaluated by summing the resulting geometric series. The common ratio $\rho$ is identified as the conjugate of the period unit $u = k_{N-1}\xi + k_{N-2}$.

For generalized continued fractions of the form $\xi = a_0 + \frac{b_1}{a_1 + \frac{b_2}{a_2 + \dots}}$, the authors utilize an algebraic telescoping argument. By defining $c_n = 1/B_n$ (where $B_n$ is the product of numerators), they construct a recurrence that allows the sum of squared errors to telescope, yielding an identity for $\xi$ under specific convergence conditions.

Key Contributions and Results

1. Geometric Decomposition of Errors: The paper establishes that for a purely periodic quadratic irrational $\xi$ with period $N$, the error sequence $\varepsilon_n$ splits into $N$ geometric subsequences. For any residue class $r \in \{-1, 0, \dots, N-2\}$, the errors satisfy:
   $$\varepsilon_{mN+r} = \varepsilon_r \cdot \rho^m$$
   where $\rho = \bar{u}$ is the conjugate of the fundamental period unit $u$. This result, described as a "simple and nice" observation not previously found in the literature, allows for the explicit calculation of error sums.

2. Explicit Closed Formulae for Weighted Sums: The authors derive a closed formula for the weighted error sum $f_\xi(s)$:
   $$f_\xi(s) = \sum_{r=-1}^{N-2} a_{r+1} \frac{|\varepsilon_r|^s}{1 - |\rho|^s}$$
   They prove that for integer $s \ge 1$, $f_\xi(s)$ belongs to the field $\mathbb{Q}(\xi)$. This provides a unified generalization of earlier identities (such as the case $s=1$ or $s=2$) for quadratic irrationals.

3. Connection to Fundamental Units: The common ratio $\rho$ is explicitly linked to the fundamental unit of the real quadratic order generated by $\xi$. The period unit $u = k_{N-1}\xi + k_{N-2}$ is shown to be the fundamental unit (or a power thereof) of the corresponding order, with norm $N(u) = (-1)^N$.

4. Generalized Identities for $\pi$ and $\ln 2$: Extending the method to generalized continued fractions, the authors prove a telescoping identity:
   $$\sum_{n=-1}^{\infty} \frac{a_{n+1}}{B_{n+1}} (h_n - \xi k_n)^2 = \xi$$
   Applying this to specific expansions:

   • For $\pi$: Using the generalized continued fraction derived from the arctangent series, they obtain an identity involving the tail of the Leibniz series and the digamma function $\psi(z)$.
   • For $\ln 2$: Using Euler's continued fraction for $\ln(1+x)$ at $x=1$, they derive a sum of squares of the tails of the alternating harmonic series.
   • Faster Convergence: They also apply the method to a series for $\log 2$ derived via Euler's transformation, yielding a faster-converging identity involving the digamma function.

Significance
The paper claims significance in providing a unified framework for understanding weighted error sums in periodic continued fractions. By identifying the underlying geometric progression of errors, the authors transform the problem of summing infinite series of approximation errors into a finite calculation involving the initial segment of the continued fraction and the period unit.

The work generalizes previous results (such as those by Kalinin and Shkolnikov) which were limited to specific cases ($s=1, 2$) or specific constants. Furthermore, the extension to generalized continued fractions allows for the derivation of new identities for fundamental constants like $\pi$ and $\ln 2$, expressed in terms of their convergent tails and special functions. The authors note that these results illustrate the "diversity and continued activity" of research in Diophantine approximation and error-sum functions, offering a structural explanation for previously observed identities.

The paper concludes by modestly suggesting future directions, such as multidimensional analogues using algorithms like Jacobi-Perron, while acknowledging the difficulty of guaranteeing periodicity in higher dimensions (Hermite's problem).