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Convergent Approximation for the Golden Tail

Abstract

The three-periodic golden continued fraction gives integral convergents with explicit approximation order and leading error coefficient.

Theorem 1.1 (The golden tail and its convergent errors).

Lean statement: D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedGolden.golden_approximation

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MetallicHankel/MetallicHankelUnboundedGolden.golden_approximation (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Guo-Niu Han, Emmanuel Pedon (2025). Hankel continued fractions and Hankel determinants for q-deformed metallic numbers. DOI: 10.48550/arXiv.2502.05993. URL: https://arxiv.org/abs/2502.05993v2.

Commentary.

For each nonnegative integer p, set k_p = 1 when p is congruent to two modulo three and k_p = 0 otherwise. Set v_0 = 1; at positive p, set v_p = 1 when p is congruent to one modulo three and v_p = -1 otherwise. Set D_p = 1+q-q^2 when p is congruent to two modulo three and D_p = 1+q otherwise. Let s_0 = 0, s_{p+1} = s_p+k_p+1, h_0 = v_0 and h_{p+1} = h_p v_{p+1}. Let Q_0 = 1, Q_1 = D_0, N_0 = 0 and N_1 = v_0 q^{k_0}. Suppose both Q and N satisfy U_{p+2} = D_{p+1} U_{p+1} - v_{p+1} q^{k_p+k_{p+1}+2} U_p over integral formal power series. There is a family F of integral formal power series such that q^3 F_0^2 + (1+q-q^2)F_0 = 1. For every p, Q_p has constant coefficient one and no coefficients above degree s_p, while N_p has no coefficients at degrees at least s_p. Moreover, for every p there is an integral formal power series R_p with Q_p F_0 - N_p = q^{2s_p+k_p} R_p and constant coefficient of R_p equal to h_p.

References