Weighted error-sum identities for periodic continued fractions and their generalizations
Kevin Calderon, Nikita Kalinin
TL;DR
This work reveals that for purely periodic continued fractions $\xi=[\overline{a_0,\dots,a_{N-1}}]$, the approximation errors split into $N$ geometric subsequences, yielding explicit closed forms for the weighted error sums $f_{\xi}(s)=\sum a_{n+1}|h_n-\xi k_n|^s$ in terms of the initial block and a common ratio $\rho\in\mathbb{Q}(\xi)$. The authors prove the geometric decay via three independent approaches (matrix dynamics, algebraic conjugation, and complete quotients) and connect $\rho$ to a real quadratic unit $u=k_{N-1}\xi+k_{N-2}$ with $u\bar{u}=(-1)^N$, embedding the phenomenon in the arithmetic of algebraic units. They extend the framework to generalized continued fractions with numerators $(b_n)$, obtaining Euler-type identities and concrete expressions for special constants such as $\pi$ and $\ln 2$, including digamma representations. The paper further discusses the role of units and Pell-type relations, and sketches multidimensional generalizations via Jacobi–Perron expansions, aiming at analogous geometric-error identities in higher degrees. Overall, the results provide a unified, algebraic view of weighted error sums for quadratic irrationals and open avenues for higher-dimensional generalizations with potential applications to number theory and Diophantine approximation.
Abstract
For a purely $N$-periodic continued fraction $ξ=[\overline{a_0,a_1,\dots,a_{N-1}}]=[a_0,a_1,\cdots]$, with $a_k=a_{k+N}$ for all $k\ge 0$, and convergents $h_n/k_n=[a_0,a_1,\dots,a_n]$, we obtain explicit expressions for the weighted error sums $f_ξ(s)=\sum a_{n+1}\lvert h_n-ξk_n\rvert^s$ for $s>1$. A key observation is that, for each residue class $k_0\in{0,1,\dots,N-1}$, the subsequence of approximation errors $(h_k-ξk_k)$ with $k\equiv k_0 \pmod N$ forms a geometric progression. In addition, we extend our methods to generalized continued fractions with numerators $(b_n)$, obtaining Euler-type identities and weighted error-sum formulae for $π$ and $\ln 2$.