Precise Asymptotics and Exact Formulas for Tensor Product Energies of Fibonacci Lattices

Abstract

We consider the asymptotics of sums of the form $$ \frac1{F_n^σ} \sum_{m = 1}^{F_n-1} \frac{f(m/F_n)}{\left|{\sin(πm/F_n)}\right|^σ} \frac{f(F_{n-1}m/F_n)}{\left|{\sin(πF_{n-1}m/F_n)}\right|^σ} $$ where $(F_n)_{n \in \mathbb N} = (1, 1, 2, 3, 5, 8, 13, \dots)$ are the Fibonacci numbers. Such sums appear, for example, in the context of discrepancy theory and numerical integration methods reformulated as energy minimization problems. We show that for parameters $σ> 1$ and a large class of functions $f$ the above sum behaves asymptotically like $$ C n + D + O\left((1-\varepsilon)^{n}\right) $$ for some constants $C$ and $D$. These constants can be given via infinite series connected to the Dedekind zeta function over the algebraic number field $\mathbb Q(\sqrt5)$. In special cases we even observe simple closed-form expressions for such sums as above, explicitly proving that $$ \sum_{m=1}^{F_n-1} \frac1{\sin(πm/F_n)^2} \frac1{\sin(πF_{n-1} m/F_n)^2} = \frac{4n}{75} F_{2n} - \frac{17}{225}F_n^2 - (-1)^n \frac2{15} - \frac19. $$