[Paper Review] Weighted error-sum identities for periodic continued fractions and their generalizations

Kevin Calderon, Nikita Kalinin|arXiv (Cornell University)|Jan 10, 2026

Mathematical functions and polynomials0 citations

TL;DR

The paper proves that for purely periodic quadratic irrationals, the error sequence in convergents splits into N geometric subsequences and derives explicit, unit-related expressions for weighted error sums; it also extends to generalized continued fractions and obtains Euler-type identities for π and ln 2.

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 $f_ξ(s)=\sum a_{n+1}\lvert h_n-ξk_n vert^s$ 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 $\ln 2$.

Motivation & Objective

• Characterize the error terms in convergents of purely periodic quadratic irrationals.
• Show that the error sequence decomposes into N geometric subsequences with a common ratio in Q(ξ).
• Obtain closed-form expressions for weighted error sums fξ(s) and relate them to algebraic units.
• Extend the framework to generalized continued fractions with Euler-type identities.
• Derive connections to fundamental units in real quadratic fields and explore multidimensional generalizations.

Proposed method

• Establish a geometric decomposition of errors εn = hn − ξkn into N subsequences εmN+r = εr ρm with ρ in Q(ξ).
• Provide three proofs: matrix dynamics using the period matrix MN; algebraic conjugation in Q(ξ) via u = kN−1 ξ + kN−2; and complete quotients of the continued fraction.
• Define βk(s) = sum over indices j ≡ k (mod N) of |hn − ξ kn|^s and show βk(s) = |εk|^s /(1 − |ρ|^s).
• Define fξ(s) = sum n≥−1 a(n+1) |hn − ξ kn|^s and express as fξ(s) = sum i=0..N−1 a_{i+1} βi(s).
• Relate ρ to the unit u = kN−1 ξ + kN−2 and derive corollaries in Q(ξ).
• Extend methodology to generalized continued fractions with numerators bn and prove an Euler-type identity for weighted error sums.

Experimental results

Research questions

• RQ1 How do the approximation errors in convergents of purely periodic quadratic irrationals distribute across residue classes modulo the period?
• RQ2 What is the explicit form of the weighted error-sum fξ(s) for integer s ≥ 1, and how is it expressed in terms of the initial segment and the geometric ratio ρ?
• RQ3 How is the common ratio ρ connected to algebraic units in the underlying quadratic field?
• RQ4 Can the framework be extended to generalized continued fractions with split denominators and yield analogous identities for π and ln 2?
• RQ5 What multidimensional analogues could generalize these identities to higher-degree or Jacobi–Perron-type expansions?

Key findings

• The error sequence for purely periodic ξ splits into N geometric subsequences with a common ratio ρ in Q(ξ).
• βk(s) sums form geometric series: βk(s) = |εk|^s /(1 − |ρ|^s).
• For integer s ≥ 1, fξ(s) lies in Q(ξ).
• ρ equals (−1)^N / u with u = kN−1 ξ + kN−2, where u is a unit in the real quadratic field K = Q(ξ).
• The unit interpretation leads to expressions of fξ(s) in terms of initial errors and the fundamental unit.
• The paper extends these ideas to generalized continued fractions, yielding Euler-type identities and further connections to π and ln 2.